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GLOBAL REGULARITY FOR 2D WATER WAVES WITH SURFACE TENSION ALEXANDRU D. IONESCU AND FABIO PUSATERI Abstract. We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of our analysis is to develop a sufficiently robust method (the “quasilinear I-method”) which allows us to deal with strong singularities arising from time resonances in the applications of the normal form method (the so-called “division problem”). As a result, we are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions. Part of our analysis relies on a new treatment of the Dirichlet-Neumann operator in di- mension two which is of independent interest. As a consequence, the results in this paper are self-contained. Contents 1. Introduction 1 2. Preliminaries 11 3. Derivation of the main scalar equation 14 4. Energy estimates I: high Sobolev estimates 23 5. Energy estimates II: low frequencies 34 6. Energy estimates III: weighted estimates for high frequencies 38 7. Energy estimates IV: weighted estimates for low frequencies 44 8. Decay estimates 50 9. Proof of Lemma 8.6 59 10. Modified scattering 72 Appendix A. Analysis of symbols 76 Appendix B. The Dirichlet-Neumann operator 78 Appendix C. Elliptic bounds 90 References 98 1. Introduction 1.1. Free boundary Euler equations and water waves. The evolution of an inviscid perfect fluid that occupies a domain Ω t R n , for n 2, at time t R, is described by the free boundary incompressible Euler equations. If v and p denote respectively the velocity and the The first author was partially supported by a Packard Fellowship and NSF Grant DMS 1265818. The second author was partially supported by a Simons Postdoctoral Fellowship and NSF Grant DMS 1265875. 1
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Page 1: GLOBAL REGULARITY FOR 2D WATER WAVES … › ~fabiop › CapRevisedFinal2.pdfGLOBAL REGULARITY FOR 2D WATER WAVES WITH SURFACE TENSION ALEXANDRU D. IONESCU AND FABIO PUSATERI Abstract.

GLOBAL REGULARITY FOR 2D WATER WAVES

WITH SURFACE TENSION

ALEXANDRU D. IONESCU AND FABIO PUSATERI

Abstract. We consider the full irrotational water waves system with surface tension and nogravity in dimension two (the capillary waves system), and prove global regularity and modifiedscattering for suitably small and localized perturbations of a flat interface. An important pointof our analysis is to develop a sufficiently robust method (the “quasilinear I-method”) whichallows us to deal with strong singularities arising from time resonances in the applications ofthe normal form method (the so-called “division problem”). As a result, we are able to considera suitable class of perturbations with finite energy, but no other momentum conditions.

Part of our analysis relies on a new treatment of the Dirichlet-Neumann operator in di-mension two which is of independent interest. As a consequence, the results in this paper areself-contained.

Contents

1. Introduction 12. Preliminaries 113. Derivation of the main scalar equation 144. Energy estimates I: high Sobolev estimates 235. Energy estimates II: low frequencies 346. Energy estimates III: weighted estimates for high frequencies 387. Energy estimates IV: weighted estimates for low frequencies 448. Decay estimates 509. Proof of Lemma 8.6 5910. Modified scattering 72Appendix A. Analysis of symbols 76Appendix B. The Dirichlet-Neumann operator 78Appendix C. Elliptic bounds 90References 98

1. Introduction

1.1. Free boundary Euler equations and water waves. The evolution of an inviscidperfect fluid that occupies a domain Ωt ⊂ Rn, for n ≥ 2, at time t ∈ R, is described by the freeboundary incompressible Euler equations. If v and p denote respectively the velocity and the

The first author was partially supported by a Packard Fellowship and NSF Grant DMS 1265818. The secondauthor was partially supported by a Simons Postdoctoral Fellowship and NSF Grant DMS 1265875.

1

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2 ALEXANDRU D. IONESCU AND FABIO PUSATERI

pressure of the fluid (with constant density equal to 1) at time t and position x ∈ Ωt, theseequations are: vt + v · ∇v = −∇p− gen x ∈ Ωt

∇ · v = 0 x ∈ Ωt

v(x, 0) = v0(x) x ∈ Ω0 ,(1.1)

where g is the gravitational constant. The free surface St := ∂Ωt moves with the normalcomponent of the velocity according to the kinematic boundary condition:

∂t + v · ∇ is tangent to⋃t

St ⊂ Rn+1 . (1.2a)

In the presence of surface tension the pressure on the interface is given by

p(x, t) = σκ(x, t) , x ∈ St , (1.2b)

where κ is the mean-curvature of St and σ > 0. At liquid-air interfaces, the surface tensionforce results from the greater attraction of water molecules to each other than to the moleculesin the air. In the case of irrotational flows, i.e.

curl v = 0 , (1.3)

one can reduce (1.1)-(1.2) to a system on the boundary. Such a reduction can be performedidentically regardless of the number of spatial dimensions, but here we only focus on the twodimensional case – which is the one we are interested in – and moreover assume that Ωt ⊂ R2 isthe region below the graph of a function h : Rx×Rt → R, that is Ωt = (x, y) ∈ R2 : y ≤ h(x, t)and St = (x, y) : y = h(x, t).

Let us denote by Φ the velocity potential: ∇Φ(x, y, t) = v(x, y, t), for (x, y) ∈ Ωt. Ifφ(x, t) := Φ(x, h(x, t), t) is the restriction of Φ to the boundary St, the equations of motionreduce to the following system for the unknowns h, φ : Rx × Rt → R:

∂th = G(h)φ

∂tφ = −gh+ σ∂2xh

(1 + h2x)3/2

− 1

2|φx|2 +

(G(h)φ+ hxφx)2

2(1 + |hx|2)

(1.4)

with

G(h) :=

√1 + |hx|2N (h) (1.5)

where N (h) is the Dirichlet-Neumann1 map associated to the domain Ωt. We refer to [46,chap. 11] or [19] for the derivation of the water waves equations (1.4). This system describesthe evolution of an incompressible perfect fluid of infinite depth and infinite extent, with afree moving (one-dimensional) surface, and a pressure boundary condition given by the Young-Laplace equation. One generally refers to (1.4) as the gravity water waves system when g > 0and σ = 0, and as the capillary water waves system when g = 0 and σ > 0.

The system (1.1)-(1.2) has been under very active investigation in recent years. Withouttrying to be exhaustive, we mention the early work on the wellposedness of the Cauchy problemin the irrotational case and with gravity by Nalimov [40], Yosihara [49], and Craig [17]; thefirst works on the wellposedness for general data in Sobolev spaces (for irrotational gravitywaves) by Wu [50, 51]; and subsequent work on the gravity problem by Christodoulou-Lindblad

1By slightly abusing notation we will refer to G(h) as the Dirichlet-Neumann map, as this causes no confusion.

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WATER WAVES WITH SURFACE TENSION 3

[12], Lannes [37], Lindblad [39], Coutand-Shkoller [15], Shatah-Zeng [44, 45], and Alazard-Burq-Zuily [2, 3]. Surface tension effects have been considered in the work of Beyer-Gunther[8], Ambrose-Masmoudi [7], Coutand-Shkoller [15], Shatah-Zeng [44, 45], Christianson-Hur-Staffilani [11], and Alazard-Burq-Zuily [1]. Recently, some blow-up scenarios have also beeninvestigated [10, 9, 16, 30].

The question of long time regularity of solutions with irrotational, small and localized initialdata was also addressed in a few papers, starting with [52], where Wu showed almost globalexistence for the gravity problem (g > 0, σ = 0) in two dimensions (1d interfaces). Subse-quently, Germain-Masmoudi-Shatah [23] and Wu [53] proved global existence of gravity wavesin three dimensions (2d interfaces). Global regularity in 3d was also proved in the case ofsurface tension and no gravity (g = 0, σ > 0) by Germain-Masmoudi-Shatah [24].

Global regularity for the gravity water waves system in dimension 2 (the harder case) hasbeen proved by the authors in [32].2 and, independently by Alazard-Delort [4, 5]. More recently,a new proof of Wu’s 2d almost global existence result was given by Hunter-Ifrim-Tataru [27],and then complemented to a proof of global regularity in [28].

After the results of this paper were completed and presented in lectures3, Ifrim-Tataru [29]announced an independent proof of a global existence result for the same system, in the caseof data that satisfies one momentum condition on the Hamiltonian variable.

1.2. The main results. Our results in this paper concern the capillary water waves system∂th = G(h)φ,

∂tφ =∂2xh

(1 + h2x)3/2

− 1

2|φx|2 +

(G(h)φ+ hxφx)2

2(1 + |hx|2).

(1.6)

This is the system (1.4) when gravity effects are neglected (g = 0) and the surface tensioncoefficient σ is, without loss of generality, taken to be 1. The system admits the conservedHamiltonian

H(h, φ) :=1

2

∫RG(h)φ · φdx+

∫R

(∂xh)2

1 +√

1 + h2x

dx ≈∥∥|∂x|1/2φ∥∥2

L2 +∥∥|∂x|h∥∥2

L2 . (1.7)

To describe our results we first introduce some basic notation. Let

C0 := f : R→ C , f continuous and lim|x|→∞

|f(x)| = 0, ‖f‖C0 := ‖f‖L∞ . (1.8)

For any N ≥ 0 let HN denote the standard Sobolev space of index N . More generally, if N ≥ 0,b ∈ [−1, N ], and f ∈ C0 then we define

‖f‖HN,b :=∑k∈Z‖Pkf‖2L2(22Nk + 22kb)

1/2≈∥∥(|∂x|N + |∂x|b)f

∥∥L2 ,

‖f‖WN,b :=∑k∈Z‖Pkf‖L∞(2Nk + 2bk),

(1.9)

2See also our earlier paper [31] for the analysis of a simplified model (a fractional cubic Schrodinger equation),and [33] for an alternative description of the asymptotic behavior of the solutions constructed in [32].

3For example at the Banff workshop “Dynamics in Geometric Dispersive Equations and the Effects of Trap-ping, Scattering and Weak Turbulence” on May 6 and 7, 2014.

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4 ALEXANDRU D. IONESCU AND FABIO PUSATERI

where Pk denote standard Littlewood-Paley projection operators (see subsection 2.1 for precise

definitions). Notice that the norms HN,b define natural spaces of distributions for b < 1/2 indimension 1, but not for b ≥ 1/2. This is the reason for the assumption f ∈ C0 in the definition.Our main result is the following:

Theorem 1.1 (Global Regularity). Let

N0 = NI := 9, N1 = NS := 3, N2 = N∞ := 5, 0 < 104p1 ≤ p0 ≤ 10−10. (1.10)

Assume that (h0, φ0) ∈ (C0 ∩ HN0+1,p1+1/2)× HN0+1/2,p1 satisfies

‖h0‖HN0+1,p1+1/2 + ‖φ0‖HN0+1/2,p1 + ‖(x∂x)h0‖HN1+1,p1+1/2 + ‖(x∂x)φ0‖HN1+1/2,p1 = ε0 ≤ ε0,(1.11)

where ε0 is a sufficiently small constant. Then, there is a unique global solution

(h, φ) ∈ C([0,∞) : (C0 ∩ HN0+1,p1+1/2)× HN0+1/2,p1

)of the system (1.6), with (h(0), φ(0)) = (h0, φ0). In addition, with S := (3/2)t∂t+x∂x, we have

〈t〉−p0‖f(t)‖HN0,p1−1/2 + 〈t〉−4p0‖Sf(t)‖HN1,p1−1/2 + 〈t〉1/2‖f(t)‖WN2,−1/10 . ε0, (1.12)

for any t ∈ [0,∞), where 〈t〉 := 1 + t and f ∈ |∂x|h, |∂x|1/2φ.

Remark 1.2 (Modified scattering). The solution exhibits modified scattering behavior ast→∞. A precise statement can be found in Theorem 10.1 in section 10, in which we providetwo different descriptions of the asymptotic behavior of the solution.

Remark 1.3 (Low frequencies and momentum conditions). The proof of the main theorembecomes easier if one makes the stronger low-frequency assumption

‖h0‖HN0+1,1/2− + ‖(x∂x)h0‖HN1+1,1/2− + ‖φ0‖HN0+1/2,0− + ‖(x∂x)φ0‖HN1+1/2,0− 1, (1.13)

which is the same condition as (1.11), but taking p1 < 0. Such low frequency norms arepropagated by the flow, due to a suitable null structure at low frequencies of the nonlinearity.However, the finiteness of the norm (1.13) requires an unwanted momentum condition on the

natural Hamiltonian variables (|∂x|h, |∂x|1/2φ), compare with (1.7).The choice of norm in (1.11) accomplishes our main goals. On one hand it is strong enough

to allow us to control the singular terms arising from the resonances of the normal form trans-formation and deal with the “division problem”, see 1.5 below. On the other hand, it is weakenough to avoid momentum assumptions on the natural energy variables4 |∂x|h and |∂x|1/2φ.

As a byproduct of our energy estimates in sections 4 and 5, we also obtain the following:

Theorem 1.4 (Long-time existence in Sobolev spaces). Assume that

(h0, φ0) ∈ (C0 ∩ HN0+1,p1+1/2)× HN0+1/2,p1

satisfies‖h0‖HN0+1,p1+1/2 + ‖φ0‖HN0+1/2,p1 = ε0 ≤ 1. (1.14)

Then there is a unique solution

(h, φ) ∈ C([0, Tε0 ] : (C0 ∩ HN0+1,p1+1/2)× HN0+1/2,p1

)4Similar assumptions at low frequencies, designed to avoid momentum conditions on the energy variables,

were used by Germain-Masmoudi-Shatah [24] in their work on the capillary system in three dimensions.

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WATER WAVES WITH SURFACE TENSION 5

of the system (1.6), with (h(0), φ(0)) = (h0, φ0) and Tε0 & ε−20 . Moreover,

‖h(t)‖HN0+1,p1+1/2 + ‖φ(t)‖HN0+1/2,p1 . ε0 (1.15)

for any t ∈ [0, Tε0 ].

A similar result also holds in the case of periodic solutions. The result in Theorem 1.4 isthe first of this type for the capillary water waves system with general initial conditions in theSchwartz class (in the sense explained in remark 1.3 above). For gravity water waves such aresult was obtained in5 [47] in 2d, and in [48] in 3d, as a key step to proving the modulationapproximation in infinite depth. By analogy, Theorem 1.4 should be considered an importantstep towards the rigorous justification of approximate models and scaling limits for the waterwaves system in the presence of surface tension. We refer the reader to [18, 42], the book [38],and references therein, for works dealing with the long-time existence of non-localized solutionsof the water waves system and their modulation and scaling regimes.

1.3. Main ideas of the proof. The system (1.6) is a time reversible quasilinear system. Inorder to prove global regularity for solutions of the Cauchy problem for this type of equations,one needs to accomplish two main tasks:

1) Propagate control of high frequencies (high order Sobolev norms);2) Prove pointwise decay of the solution over time.

In this paper we use a combination of improved energy estimates and asymptotic analysis toachieve these two goals. This is a natural continuation of our work on the gravity water wavessystem [32]. However, here we adopt a more robust framework for most of the arguments.Unlike our previous work [32] where we used both Lagrangian and Eulerian coordinates, herewe perform our proof entirely in Eulerian coordinates6. We carry out both main parts of ouranalysis in Fourier space. More precisely, we use a “quasilinear I-method”, originally introducedin our recent work [34] on a simplified model (see also [21] for a different application of themethod in a 2d problem), to construct high order energy functionals which can be controlledfor long times. In order to set up the equations for the energy estimates, we perform a carefulparalinearization in the spirit of [1, 5], but better adapted for our purposes. An importantpart of the argument here relies on new bounds on the Dirichlet-Neumann operator in twodimensions, consistent with the limited low-frequency structure we assume on the interface h,see (1.11). To complete our proof we then use the Fourier transform method to obtain sharptime decay rates for our solutions, and prove modified scattering. We elaborate on all thesemain aspects of our proof below.

1.4. Paralinearization and the Dirichlet-Neumann operator. One of the main difficul-ties in the analysis of the water waves system (1.4) comes from the quasilinear nature of theequations, and their non-locality, due to the presence of the Dirichlet-Neumann operator G(h)φ,see (1.5). In a series of papers [1, 2, 3] Alazard-Burq-Zuily proposed a systematic approachto these issues in the context of the local-in-time Cauchy problem, based on para-differentialcalculus. See also the earlier works of Alazard-Metivier [6] and Lannes [37]. This approachwas then extended and adapted to study the problem of global regularity for gravity waves byAlazard-Delort in [5].

5It is also a corollary of the results in [52, 32, 5, 27].6Besides being the natural coordinates associated to the Hamiltonian formulation of the water waves problem,

Eulerian coordinates are in general more flexible because the analysis can be generalized to higher dimensions.

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6 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Our first step towards the proof of Theorem 1.1 is a paralinearization of the system (1.6)inspired by the works cited above. For our problem, however, we need bounds that dependonly on ‖|∂x|1/2+p1h‖L2 , not on ‖h‖L2 . More precisely, let

(h, φ) ∈ C(C0 ∩ HN0+1,p1+1/2(R)×HN0+1/2,p1(R)) (1.16)

be a real-valued solution of (1.6) on some time interval. Define

B :=G(h)φ+ hxφx

(1 + h2x)

, V := φx −Bhx, ω := φ− TBP≥1h. (1.17)

Here, for any a, b ∈ L2(R), we have denoted by Tab the paradifferential operator

F(Tab)(ξ) :=1

∫Ra(ξ − η)b(η)χ0(ξ − η, η) dη,

χ0(x, y) :=∑k∈Z

ϕ≤k−10(x)ϕk(y),(1.18)

which restricts the product of a and b to the frequencies of b much larger than those of a. Thefunctions (V,B) represent the restriction of the velocity field v to the boundary of Ωt, and thefunction ω is a variant of the so-called “good-unknown” of Alinhac.

In Appendix B, we prove the following formula for the Dirichlet-Neumann operator:

G(h)φ = |∂x|ω − ∂xTV h+G2 +G≥3, (1.19)

where G2 is an explicit semilinear quadratic term, and G≥3 denotes cubic and higher orderterms. This formula was already derived and used by Alazard-Delort [5]; the new aspect hereis that the remainder G≥3 satisfies better trilinear bounds of L2, weighted L2, and L∞-type,

under the sole assumptions (1.16). In particular we only need to assume that |∂x|1/2+p1h and|∂x|p1φ are in L2, for some p1 > 0.

In section 3, using (1.19), we diagonalize and symmetrize the system (1.6) reducing it to asingle scalar equation for one complex unknown u of the form

u ≈ |∂x|h− i|∂x|1/2ω + higher order corrections, (1.20)

see the formulas (3.16)-(3.17). The capillary system (1.6) then takes the form

∂tu− iΛ(∂x)u− iΣ(u) = −TV ∂xu+Nu +Ru, (1.21)

where: Λ(ξ) := |ξ|3/2 is the dispersion relation for linear waves of frequency ξ; Σ can bethought of as a self-adjoint differential operator of order 3/2 which is cubic in u; Nu aresemilinear quadratic terms; and Ru contains (semilinear) cubic and higher order terms in u.See Proposition 3.2 for details. Equation (1.21) is our starting point in establishing energy

estimates for u, hence for (|∂x|h, |∂x|1/2φ).

1.5. The “division problem” and energy estimates via a “quasilinear I-method”.In order to construct high order energy functionals controlling Sobolev norms of our solution,we first apply the natural differentiation operator associated to the equation (1.21). More

precisely, we look at Wk := Dku, with D := |∂x|3/2 + Σ, and show that

∂tWk − iΛ(∂x)Wk − iΣ(Wk) = −TV ∂xWk +NWk+RWk

. (1.22)

Here Wk ≈ |∂x|3k/2u, NWkdenotes semilinear quadratic terms, and RWk

are semilinear cubic

and higher order terms in Wk. If one looks at the basic functional E(t) = ‖WN0(t)‖2L2 associated

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WATER WAVES WITH SURFACE TENSION 7

to (1.22), it is easy to verify that, as long as solutions are of size ε, such energy functional iscontrolled for O(ε−1) times.

In order to go past this local existence time, one needs to rely on the dispersive properties ofsolutions. One of the main difficulties in dealing with a one dimensional problem such as (1.22)

is the slow time decay, which is t−1/2 for linear solutions. A classical idea used to overcome thedifficulties associated to weak dispersion is the use of normal forms [43], which can sometimesbe used to eliminate the slow decaying quadratic terms from the nonlinearity.

In the context of water waves in 2D (1D interfaces), the starting point is an estimate on theenergy increment of the form

∂tE(t) = quartic semilinear terms,

where E is a suitable energy functional. An energy inequality of this form was first proved byWu [52] for the gravity water wave model, and led to an almost-global existence result. Allthe later work on long term 2D water wave models, such as [32, 34, 4, 5, 27, 28, 29] and thispaper, relies on proving an inequality of this type as a first step. Such an inequality (which isrelated to normal form transformations) is at least formally possible when there are no timeresonances.

For (1.22) a normal form transformation is formally available since the only time resonances,i.e. solutions of

Λ(ξ)± Λ(ξ − η)± Λ(η) = 0, (1.23)

occur when one of the three interacting frequencies (ξ, ξ−η, η) is zero. However, the superlinear

dispersion relation |ξ|3/2 makes these resonances very strong. For example, in the case |ξ| ≈|η| ≈ 1 |ξ − η|, we see that

|Λ(ξ)± Λ(ξ − η)− Λ(η)| ≈ |ξ − η|. (1.24)

Because of (1.23), a standard normal transformation which eliminates the quadratic terms in(1.22) will introduce a low frequency singularity. This is the “division problem” mentioned

earlier. For comparison, in the gravity water waves case the dispersion relation is Λ(ξ) = |ξ|1/2and one has the less singular behavior |Λ(ξ) ± Λ(ξ − η) − Λ(η)| ≈ |ξ − η|1/2 in the case|ξ| ≈ |η| ≈ 1 |ξ − η|.

The implementation of the method of normal forms is delicate in quasilinear problems, dueto the potential loss of derivatives. Nevertheless this has been done in some cases, for exampleby using carefully constructed nonlinear changes of variables as in Wu [52], or via the “iteratedenergy method” of Germain-Masmoudi [22], or the “paradifferential normal form method” ofAlazard-Delort [5], or the “modified energy method” of Hunter-Ifrim-Tataru [27].

In order to deal with the issues of slow decay and simultaneous strong time resonances, wepropose a Fourier based approach to normal forms performed at the level of energy functionals,which we implement in a similar spirit to the I-method of Colliander-Keel-Staffilani-Takaoka-Tao [13, 14].

A related construction in the physical space and in holomorphic coordinates was performedin [27] in the case of gravity water waves. In the gravity case, however, the problem is simplerbecause there are no singularities in the resulting quartic integrals. In our case, however, thereare strong singularities (of the form (low frequency)−1/2) in the quartic integrals resulting fromthe application of the normal form. To deal with these singularities it is important to construct

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8 ALEXANDRU D. IONESCU AND FABIO PUSATERI

energy functionals in the Fourier space (the so-called I-method) and make careful assumptionson the low frequency structure of solutions.

We begin our analysis by looking the basic energy, expressed in time-frequency space,

E(2)(t) =1

∫R|WN0(ξ, t)|

2dξ +

1

∫R|SWN1(ξ, t)|

2dξ, (1.25)

forN0 andN1 as in (1.10). Assuming that E(2)(0) is of size ε20, our maim aim is to control E(2)(t)

for all times, possibly allowing a small polynomial growth, under the a priori assumption thatsolutions are ε0t

−1/2 small in a suitable L∞x type space. As we will see below, this is not possiblewithout stronger information on the low-frequencies behavior of the solution. Calculating theevolution of (1.25) by using the equations (1.22), and performing appropriate symmetrizationsto avoid losses of derivatives, we obtain

∂tE(2)(t) = semilinear cubic terms. (1.26)

We then define a cubic energy functional E(3) which is a sum of cubic terms of the form∫R×R

F (ξ, t)G(η, t)H(ξ − η, t)m(ξ, η) dξdη (1.27)

where F,G,H can be any of the functions WN0 , SWN1 , u, Su or their complex conjugates, andthe symbols m are obtained by diving the symbols of the cubic expressions in (1.26) by theappropriate resonant phase function (1.23). These cubic energy functionals are a perturbationof (1.25) on each fixed time slice. Moreover, by construction

∂t(E(2) + E(3)

)(t) = singular quartic semilinear terms. (1.28)

The “division problem” mentioned above, see (1.23)-(1.24), manifests itself in the fact that some

of the symbols of the above quartic expressions have singularities of the form (low frequency)−1/2.This is ultimately due to the lack of symmetries in the equation for Su (or SWN1).

A typical example of a singular quartic term can be schematically written (in real space, forsimplicity of exposition) as ∫

RSu · u · |∂x|1/2u · |∂x|−1/2Sudx. (1.29)

The difficulty in estimating such a term is the following: in order to control the growth of theenergy, one would need to bound (1.29) by ε0t

−1E(2)(t). A sharp a priori decay assumption

gives us that ‖u · |∂x|1/2u‖L∞ . ε20t−1, and the energy controls ‖Su‖L2 .

√E(2), but we have

no control over the L2 norm of |∂x|−1/2Su. This difficulty is not present in the gravity waterwave system, due to the less singular denominators.

To deal with these singularities we need to need to make our suitable low frequency assump-tions (1.11) on the solutions. Using these low frequency assumptions, and anticipating a sharp

decay rate of t−1/2 for our solution, we then control the energy functional E(2)(t) for all times,allowing a slow growth of t2p0 (in fact, due to weaker symmetries, we need a faster growth rateon the weighted energies). This is done in sections 4 and 6, for the Sobolev and the weightedSobolev norm.

The additional assumptions made in order to close the energy estimates above are then re-covered by low frequency energy estimates in sections 5 and 7. By following a similar argument

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WATER WAVES WITH SURFACE TENSION 9

to the one above, we control for long times a quadratic energy of the form

E(2)low :=

∫R

(|u(ξ, t)|2 + |Su(ξ, t)|2

)· |ξ|−1Pt(ξ) dξ, (1.30)

where Pt : ξ ∈ R→ [0, 1] is a smooth function that vanishes if |ξ| ≥ 20, equals 1 if (1 + t)−2 ≤|ξ| ≤ 1, and is equal to [(1 + t)2|ξ|]2p1 , if 2|ξ| ≤ (1 + t)−2. Notice that E

(2)low has the following

properties:

(1) it is consistent with our initial assumptions (1.11);

(2) for frequencies that are not too small with respect to time, it controls |∂x|−1/2Su, which isexactly what would be needed to bound the quartic singular term (1.29) above;

(3) for very small frequencies it controls the L2 norm of |∂x|−1/2+p1Su with a bound thatimproves as t→∞.

To control the growth of E(2)low(t) we follow a similar strategy to the one above and construct

a suitable cubic correction E(3)low such that, schematically, we obtain

∂t(E

(2)low + E

(3)low

)(t) = special cubic semilinear terms + singular quartic semilinear terms.

The cubic terms in the above right-hand side are special because they are supported on smalltime-dependent sets. The key to control these low frequency energy functionals is the nullstructure for low frequency outputs in the water waves system (1.6). In essence, here weexploit the fact that the nonlinear part of the system has a better low frequency behavior thanthe linear evolution, so that, without imposing moment conditions on the initial data, we canstill recover strong enough low frequency information for the nonlinear evolution.

1.6. Compatible vector-field structures. As in other quasilinear problems, to prove globalregularity we propagate control not only of high Sobolev norms but also of other L2 normsdefined by vector-fields [36]. These norms are helpful both in proving decay and in controllingremainders from the stationary phase and integration by parts arguments in section 8 and 9.In our case, a natural vector-field to propagate is the scaling vector-field S = (3/2)t∂t + x∂xwhich (essentially) commutes with the linear part of the equation.

In the context of the quasilinear I-method, propagating L2 control of vector-fields carryingweights is challenging. The reason for this is simple: as in the semilinear case, the success ofthe I-method ultimately depends on exploiting certain symmetries of the equation, which arerelated to its Hamiltonian structure. Once weighted vector-fields, such as S, are applied to theequation, these symmetries are weakened. Moreover, every weighted vector-field requires itsown energy functional, and its own set of cubic corrections. At the very least, this increasesconsiderably the amount of work needed to prove weighted energy estimates.

In this paper we are able to use energy estimates to propagate control of a specific compatiblevector-field structure, namely the vector-fields

∂m0x , S∂m1

x , m0 ∈ 0, . . . , N0, m1 ∈ 0, . . . , N1. (1.31)

We also propagate control of a suitable low-frequency structure described by an energy func-tional as in (1.30).

The compatible vector-field structure (1.31), using at most one weighted vector-field S andmany vector-fields ∂x, was introduced, in the setting of water waves, by the authors in [32].

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10 ALEXANDRU D. IONESCU AND FABIO PUSATERI

It was then used, and played a critical role, in all the later papers on the subject such as[27, 28, 34, 29]. The point of this choice is that:

(i) The compatible vector-field structure (1.31) can be propagated in time, with small tε lossand reasonably manageable computations;

(ii) It is still strong enough to provide almost optimal t−1/2 dispersive decay, since we work indimension one (see Lemmas 2.2 and 2.3).

1.7. Decay and modified scattering. Having established the L2 bounds described above,we eventually prove a pointwise sharp decay rate of t−1/2 for our solutions. This is done insections 8 and 9 where we follow a similar strategy as in our previous works [31, 32, 34], andstudy the nonlinear oscillations in the spirit of the ”method of space-time resonances”, as in[23, 24, 25, 31].

Our starting point is again the main paralinearized equation (1.21). As a first step we performa standard normal form transformation, i.e. a bilinear change of variables v = u+B(u, u), toremove the slowly decaying quadratic terms from the equation, see (8.10)-(8.11). In this partof the argument we do not need to pay attention to losses of derivatives, but only keep trackof the singularities introduced by the normal forms. It then suffices to prove decay for the newunknown v using the equation containing only cubic (and higher order) terms. To do this wewrite Duhamel’s formula in terms of the linear profile f = e−itΛv and study the oscillations intime and space of the resulting integral. In particular we need to deal with trilinear expressionsof the form ∫ t2

t1

∫R×R

eit(−|ξ|3/2±|ξ−η|3/2±|η−σ|3/2±|σ|3/2)

× c±±±(ξ, η, σ)f±(ξ − η, s)f±(η − σ, s)f±(σ, s) dηdσ ds,

(1.32)

where f+ = f , f− = f , cfr. (8.54). We remark once again that the symbols in the above

expressions have singularities of the form (low frequency)−1/2, see (9.11).We proceed by splitting the integrals (1.32) into several types of interactions, depending on

the size of the frequencies of the inputs relative to each other and to time, and estimate allthe different contributions in several Lemmas in sections 9.1 and 9.2. The main contributionto the integrals (1.32) comes from space-time resonances, which occur when the size of thefrequencies of the three functions is comparable to the size of the output frequency |ξ|, and thethree inputs are f, f and f .

In this case, a stationary phase analysis argument reveals that a correction to the asymptoticbehavior is needed, similarly to the case of gravity waves [32]. We refer the reader to [26, 20,35, 32, 41] and references therein, for more works related to modified scattering in dispersiveequations. We take this into account by defining a proper norm, the Z-norm in (8.32), which

essentially measures f in L∞. Using also the carefully chosen L2-norms (1.11), we are able tocontrol uniformly, over time and frequencies, the Z-norm. This gives us the desired pointwisedecay at the sharp t−1/2 rate, as well as modified scattering.

Some of the proofs in section 9 are similar to parts of the arguments in our previous works[31, 32, 34]. However, the decay analysis here is more complicated, once again, because of thelow frequency singularities introduced by the strong quadratic time resonances. This is espe-cially evident in this part of the argument where non L2-based norms need to be estimated,and meaningful symmetrizations cannot be performed. In particular, we need to deal with

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WATER WAVES WITH SURFACE TENSION 11

singularities in several places: to prove bounds on the normal form that recasts the quadraticnonlinearity into a cubic one (subsection 8.2); to control all cubic terms once the main asymp-totic contribution is factored out (subsection 9.1); to estimate the quartic terms arising fromthe renormalization of the cubic equation needed to correct the asymptotic behavior (sub-section 9.3). Moreover, the superlinear dispersion relation creates additional cubic resonantinteractions (subsection 9.2) which are not present in the case of gravity water waves.

1.8. Organization. The rest of the paper is organized as follows: in section 2 we summarizethe main definitions and notation in the paper and state the main bootstrap proposition, whichis Proposition 2.4.

In sections 3–7 we prove the main improved energy estimate. The key components of theproof are Proposition 3.2 (derivation of the main quasilinear scalar equation), Proposition 4.1(improved energy estimate on the high Sobolev norm), Proposition 5.1 (improved energy esti-mate on the low frequencies), Proposition 6.1 (improved weighted energy estimate on the highfrequencies), and Proposition 7.1 (improved weighted energy estimate on the low frequencies).The proofs in these sections use also the material presented in the appendices, in particularthe paralinearization of the Dirichlet–Neumann operator in Proposition B.1.

In sections 8–9 we prove the main improved decay estimate. In these sections we work withthe Eulerian variables. The key components of the proof are the normal form transformationin (8.10), the construction of the profile in (8.22) and its renormalization in (8.57), and theimproved control of Z norm in Lemma 8.6.

In section 10 we discuss the asymptotic behavior of nonlinear solutions, and provide twodescriptions of the modified scattering, one in the Fourier space and one in the physical space.

2. Preliminaries

2.1. Notation and basic lemmas. In this subsection we summarize some of our main no-tation and recall several basic formulas and estimates. We fix an even smooth functionϕ : R → [0, 1] supported in [−8/5, 8/5] and equal to 1 in [−5/4, 5/4], and define, for anyk ∈ Z,

ϕk(x) := ϕ(x/2k)− ϕ(x/2k−1), ϕ≤k(x) := ϕ(x/2k), ϕ≥k(x) := 1− ϕ(x/2k−1).

For k ∈ Z we denote by Pk, P≤k, and P≥k the operators defined by the Fourier multipliers ϕk,ϕ≤k, and ϕ≥k respectively. Moreover, let

P ′k := Pk−1 + Pk + Pk+1 and ϕ′k := ϕk−1 + ϕk + ϕk+1. (2.1)

Given s ≥ 0 let Hs denote the usual space of Sobolev functions on R. Recall the space C0

defined in (1.8). We use 3 other main norms: assume N ≥ 0, b ∈ [−1, N ], and f ∈ C0 then

‖f‖HN,b :=∑k∈Z‖Pkf‖2L2(22Nk + 22kb)

1/2,

‖f‖WN,b :=∑k∈Z‖Pkf‖L∞(2Nk + 2bk),

‖f‖WN := ‖f‖L∞ +

∑k≥0

2Nk‖Pkf‖L∞ .

(2.2)

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12 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Notice that ‖f‖HN,0 ≈ ‖f‖HN and ‖f‖WN . ‖f‖WN,0 . The spaces WN are often used in

connection with Lemma B.2 to prove iterative bounds on products of functions.

2.1.1. Multipliers and associated operators. We will often work with multipliers m : R2 → Cor m : R3 → C, and operators defined by such multipliers. We define the class of symbols

S∞ := m : Rd → C : m continuous and ‖m‖S∞ := ‖F−1(m)‖L1 <∞. (2.3)

Given a suitable symbol m we define the associated bilinear operator M by

F[M(f, g)

](ξ) =

1

∫Rm(ξ, η)f(ξ − η)g(η) dη, (2.4)

We often use the identity

SM(f, g) = M(Sf, g) +M(f, Sg) + M(f, g) (2.5)

where S = (3/2)t∂t + x∂x, f, g are suitable functions defined on I × R, and the symbol of the

bilinear operator M is given by

m(ξ, η) = −(ξ∂ξ + η∂η)m(ξ, η). (2.6)

This follows by direct calculations and integration by parts.Lemma 2.1 below summarizes some properties of symbols and associated operators (see [32,

Lemma 5.2] for the proof).

Lemma 2.1. (i) We have S∞ → L∞(R× R). If m,m′ ∈ S∞ then m ·m′ ∈ S∞ and

‖m ·m′‖S∞ . ‖m‖S∞‖m′‖S∞ . (2.7)

Moreover, if m ∈ S∞, A : R2 → R2 is a linear transformation, v ∈ R2, and mA,v(ξ, η) :=m(A(ξ, η) + v) then

‖mA,v‖S∞ = ‖m‖S∞ . (2.8)

(ii) Assume p, q, r ∈ [1,∞] satisfy 1/p+ 1/q = 1/r, and m ∈ S∞. Then, for any f, g ∈ L2(R),

‖M(f, g)‖Lr . ‖m‖S∞‖f‖Lp‖g‖Lq . (2.9)

In particular, if 1/p+ 1/q + 1/r = 1,∣∣∣ ∫R2

m(ξ, η)f(ξ)g(η)h(−ξ − η) dξdη∣∣∣ . ‖m‖S∞‖f‖Lp‖g‖Lq‖h‖Lr . (2.10)

(iii) If p1, p2, p3, p4 ∈ [1,∞] are exponents that satisfy 1/p1 + 1/p2 + 1/p3 + 1/p4 = 1 then∣∣∣ ∫R3

f1(ξ)f2(η)f3(ρ− ξ)f4(−ρ− η)m(ξ, η, ρ) dξdρdη∣∣∣

. ‖f1‖Lp1‖f2‖Lp2‖f3‖Lp3‖f4‖Lp4‖F−1m‖L1 .

(2.11)

Given any multiplier m : Rd → C, d ∈ 2, 3, and any k, k1, k2, k3, k4 ∈ Z, we define

mk,k1,k2(ξ, η) := m(ξ, η) · ϕk(ξ)ϕk1(ξ − η)ϕk2(η),

mk1,k2,k3,k4(ξ, η, ρ) := m(ξ, η, ρ) · ϕk1(ξ)ϕk2(η)ϕk3(ρ− ξ)ϕk4(−ρ− η).(2.12)

Let

X := (k, k1, k2) ∈ Z3 : max(k, k1, k2)−med(k, k1, k2) ≤ 4,

Y := (k1, k2, k3, k4) ∈ Z4 : 2k1 + 2k2 + 2k3 + 2k4 ≥ (1 + 2−10)2max(k1,k2,k3,k4),(2.13)

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WATER WAVES WITH SURFACE TENSION 13

and notice that mk,k1,k2 ≡ 0 unless (k, k1, k2) ∈ X , and mk1,k2,k3,k4 ≡ 0 unless (k, k1, k2) ∈ Y.Moreover, we will often use the notation

m(ξ, η) = O(f(|ξ|, |ξ − η|, |η|)

)⇐⇒ ‖mk,k1,k2(ξ, η)‖S∞ . f(2k, 2k1 , 2k2)1X (k, k1, k2) (2.14)

So, for example,

m(ξ, η) = O(|ξ − η|3/2

)means ‖mk,k1,k2(ξ, η)‖S∞ . 23k1/21X (k, k1, k2).

We use a similar notation for symbols of three variables,

m(ξ, η, ρ) = O(f(|ξ|, |η|, |ρ− ξ|, |ρ+ η|)

)⇐⇒

∥∥F−1(mk1,k2,k3,k4

)∥∥L1 . f(2k1 , 2k2 , 2k3 , 2k4)1Y(k1, k2, k3, k4).

(2.15)

2.1.2. Paraproducts. For any a, b ∈ L2(R) we define the paraproduct Tab by the formula

F(Tab)(ξ) :=1

∫Ra(ξ − η)b(η)χ(ξ − η, η) dη,

χ(x, y) :=∑k∈Z

ϕk(y)ϕ≤k−10(x).(2.16)

We also use the general formulas

ab = Tab+ Tba+R(a, b), F (a) = TF ′(a)a+RF (a), (2.17)

where R(a, b) and RF (a) are (substantially) more smooth remainders. The precise bounds onthe remainders R(a, b) and RF (a) depend on the context.

2.1.3. A dispersive estimate and an interpolation lemma. The following is our main lineardispersive estimate, which we use to control pointwise decay of solutions.

Lemma 2.2. For any t ∈ R \ 0, k ∈ Z, and f ∈ L2(R) we have

‖eitΛPkf‖L∞ . |t|−1/22k/4‖f‖L∞ + |t|−3/52−2k/5[2k‖∂f‖L2 + ‖f‖L2

](2.18)

and‖eitΛPkf‖L∞ . |t|−1/22k/4‖f‖L1 . (2.19)

A more precise version is proved in Lemma 10.2 in section 10. See also [34, Lemma A.1].We also use the following simple interpolation lemma (see also Lemma A.2 in [34]).

Lemma 2.3. For any k ∈ Z, and f ∈ L2(R) we have∥∥Pkf∥∥2

L∞.∥∥Pkf∥∥2

L1 . 2−k‖f‖L2

[2k‖∂f‖L2 + ‖f‖L2

]. (2.20)

Proof. By scale invariance we may assume that k = 0. It suffices to prove that∥∥P0f∥∥2

L1 . ‖f‖L2

[‖∂f‖L2 + ‖f‖L2

]. (2.21)

For R ≥ 1 we estimate∥∥P0f∥∥L1 .

∫|x|≤R

|P0f(x)| dx+

∫|x|≥R

|xP0f(x)| · 1

|x|dx

. R1/2‖P0f‖L2 +R−1/2‖xP0f(x)‖L2x

. R1/2‖f‖L2 +R−1/2[‖∂f‖L2 + ‖f‖L2

].

The desired estimate (2.21) follows by choosing R suitably.

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14 ALEXANDRU D. IONESCU AND FABIO PUSATERI

2.2. The main proposition. Given p1 ∈ [0, 10−3] we fix P = Pp1 : [0,∞)→ [0, 1] an increas-ing function, smooth on (0,∞), such that

P(x) = x2p1 if x ≤ 1/2, P(x) = 1 if x ≥ 1, xP ′(x) ≤ 10p1P(x). (2.22)

Our main theorem follows using the local existence theory and a continuity argument fromthe following main proposition:

Proposition 2.4 (Main bootstrap). Assume that

N0 = NI := 9, N1 = NS := 3, N2 = N∞ := 5, 0 < 104p1 ≤ p0 ≤ 10−10,

0 < ε0 ≤ ε1 ≤ ε2/30 1.

(2.23)

Assume T ≥ 1 and (h, φ) ∈ C([0,∞) : (C0 ∩ HN0+1,p1+1/2) × HN0+1/2,p1

)is a real-valued

solution of the system (1.6),

∂th = G(h)φ, ∂tφ =∂2xh

(1 + h2x)3/2

− 1

2φ2x +

(G(h)φ+ hxφx)2

2(1 + h2x)

. (2.24)

Let U := |∂x|h− i|∂x|1/2φ and assume that, for any t ∈ [0, T ],

〈t〉−p0KI(t) + 〈t〉−4p0KS(t) + 〈t〉1/2∑k∈Z

(2N2k + 2−k/10)∥∥PkU(t)

∥∥L∞≤ ε1, (2.25)

where 〈t〉 = 1 + t and, for O ∈ I, S,[KO(t)

]2:=

∫R|OU(ξ, t)|2 · (|ξ|−1 + |ξ|2NO)P((1 + t)2|ξ|) dξ. (2.26)

Assume also that the initial data U(0) satisfy the stronger bounds∑O∈I,S

∥∥(|∂x|−1/2+p1 + |∂x|NO)(OU)(0)∥∥L2 ≤ ε0. (2.27)

Then we have the improved bound, for any t ∈ [0, T ],

〈t〉−p0KI(t) + 〈t〉−4p0KS(t) + 〈t〉1/2∑k∈Z

(2N2k + 2−k/10)∥∥PkU(t)

∥∥L∞. ε0. (2.28)

The rest of the paper is concerned with the proof of Proposition 2.4. We will always workunder the assumptions (2.25)-(2.26). The proof depends on the equations derived in section3 and on the improved estimates in Propositions 4.1, 5.1, 6.1, 7.1, and 8.1. The argument isprovided after the statement of Proposition 8.1.

3. Derivation of the main scalar equation

As in Proposition 2.4, assume T ≥ 1 and (h, φ) ∈ C([0, T ] : HN0+1 × Hp1,N0+1/2

)is a

real-valued solution of the system (2.24) satisfying (2.25). Let

B :=G(h)φ+ hxφx

1 + h2x

, V := φx −Bhx,

ω := φ− TBP≥1h = φ− TBh+ TBP≤0h,

σ := (1 + h2x)−3/2 − 1,

(3.1)

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WATER WAVES WITH SURFACE TENSION 15

where T is defined in (2.16). Using (3.1) we calculate

− 1

2φ2x +

(G(h)φ+ hxφx)2

2(1 + h2x)

=1

2

[B2(1 + h2

x)− (V +Bhx)2]

=1

2

[B2 − 2V Bhx − V 2

]. (3.2)

Moreover, using the formula in the second line of (2.17) and standard paradifferential calculus,

∂2xh

(1 + h2x)3/2

= ∂2xh+ ∂x

( ∂xh

(1 + h2x)1/2

− hx)

= ∂2xh+ ∂xTσhx + ∂xE≥3,h, (3.3)

where E≥3,h is a more smooth cubic error (compare with (2.17)), satisfying, for any t ∈ [0, T ],

〈t〉1−p0‖E≥3,h(t)‖HN0+2 + 〈t〉1−4p0‖SE≥3,h(t)‖HN1+2 + 〈t〉11/10‖E≥3,h(t)‖WN2+2 . ε

31. (3.4)

We will also use the formula (see Proposition B.1)

G(h)φ = |∂x|ω − |∂x|TBP≤0h− ∂xTV h+G2 +G≥3, (3.5)

where

G2 := |∂x|T|∂x|φh− |∂x|(h|∂x|φ) + ∂xT∂xφh− ∂x(h∂xφ), (3.6)

and G≥3 is a cubic error, satisfying, for any t ∈ [0, T ],

〈t〉1−p0‖G≥3(t)‖HN0+1 + 〈t〉1−4p0‖SG≥3(t)‖HN1+1 + 〈t〉11/10‖G≥3(t)‖WN2+1 . ε

31. (3.7)

The function G(h)φ satisfies linear estimates with derivative loss (see (C.18) for a strongerbound)

〈t〉−p0∥∥G(h)φ

∥∥HN0−1 + 〈t〉−4p0

∥∥SG(h)φ∥∥HN1−1 + 〈t〉1/2

∥∥G(h)φ∥∥WN2−1 . ε1. (3.8)

For simplicity of notation, for α ∈ [−2, 2] let O3,α denote generic functions F on [0, T ] thatsatisfy the “cubic” bounds (see also Definition C.1)

〈t〉1−p0‖F (t)‖HN0+α + 〈t〉1−4p0‖SF (t)‖HN1+α + 〈t〉11/10‖F (t)‖WN2+α

. ε31. (3.9)

In this section we proceed with the formal calculations, without proving that the various cubicerrors terms that will appear satisfy indeed the desired bounds. All the claimed cubic boundswill follow from the assumptions (2.25) and the definitions, by elliptic estimates. Detailedproofs are provided in Appendix C.

The first equation in (2.24) becomes

∂th = |∂x|ω − |∂x|T|∂x|ωP≤0h− ∂xTV h+G2 +G≥3,

with G≥3 ∈ O3,1, while the second equation in (2.24) gives

∂tω =∂2xh

(1 + h2x)3/2

+(∂tφ−

∂2xh

(1 + h2x)3/2

)− T∂tBP≥1h− TB∂tP≥1h

=∂2xh

(1 + h2x)3/2

+ I + II + III,

(3.10)

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16 ALEXANDRU D. IONESCU AND FABIO PUSATERI

where

I = (B2 − V 2)/2− V Bhx= TBB +R(B,B)/2− TV V −R(V, V )/2− TVBhx − TBhxV +O3,1/2

= TBB − TBhxV − TV φx +R(|∂x|ω, |∂x|ω)/2−R(∂xω, ∂xω)/2 +O3,1/2,

II = T|∂x|3hP≥1h+O3,1/2,

III = −TB(G(h)φ) + TBP≤0(G(h)φ) = −TB(G(h)φ) + T|∂x|ωP≤0|∂x|ω +O3,1/2.

Notice thatB −G(h)φ− V hx = B −G(h)φ− φxhx +Bh2

x = 0.

Therefore

I + III = TB(V hx)− TBhxV − TV φx+ T|∂x|ωP≤0|∂x|ω +R(|∂x|ω, |∂x|ω)/2−R(∂xω, ∂xω)/2 +O3,1/2

= TB(V hx)− TBhxV − TV (∂xTBP≥1h)− TV ωx+ T|∂x|ωP≤0|∂x|ω +R(|∂x|ω, |∂x|ω)/2−R(∂xω, ∂xω)/2 +O3,1/2.

We show in Proposition C.3 that

TB(V hx)− TBhxV − TV (∂xTBP≥1h) ∈ O3,1/2.

Therefore, using also (3.3), the system (2.24) becomes∂th = |∂x|ω − |∂x|T|∂x|ωP≤0h− ∂xTV h+G2 +G≥3,

∂tω = ∂2xh+ ∂xTσhx − TV ωx +H2 + T|∂x|3hP≥1h+ Ω≥3,

(3.11)

where G2 := |∂x|T|∂x|φh− |∂x|(h|∂x|φ) + ∂xT∂xφh− ∂x(h∂xφ),

H2 := T|∂x|ωP≤0|∂x|ω +R(|∂x|ω, |∂x|ω)/2−R(∂xω, ∂xω)/2,

(3.12)

and, as proved in Proposition C.3,

G≥3 ∈ O3,1, Ω≥3 ∈ O3,1/2.

Let χ(x, y) := 1− χ(x, y)− χ(y, x),

m2(ξ, η) := χ(ξ − η, η)[ξ(ξ − η)− |ξ||ξ − η|]− χ(ξ − η, η)|ξ||ξ − η|ϕ≤0(η),

q2(ξ, η) := χ(ξ − η, η)[η(ξ − η) + |η||ξ − η|]/2 + χ(ξ − η, η)|η||ξ − η|ϕ≤0(η),(3.13)

and recall the notation (2.4). To summarize, we proved the following:

Proposition 3.1. Let (h, φ) be a solution of (2.24) satisfying the bootstrap assumption (2.25),and let ω, σ, V be as in (3.1). Then

∂th = |∂x|ω − ∂xTV h+M2(ω, h) +G≥3,

∂tω = ∂2xh+ ∂xTσhx − TV ωx + T|∂x|3hP≥1h+Q2(ω, ω) + Ω≥3,

(3.14)

where M2 and Q2 are the operators associated to the multipliers m2 and q2 in (3.13), and

G≥3 ∈ O3,1, Ω≥3 ∈ O3,1/2. (3.15)

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WATER WAVES WITH SURFACE TENSION 17

3.1. Symmetrization of the equations. Recall the water waves system (3.14) for the surfaceelevation h and Alinhac’s good unknown ω, and the definitions (3.1). In this section we aim todiagonalize and symmetrize this system, and write it as a single scalar equation for a complexvalued unknown u. The main result can be summarized as follows:

Proposition 3.2. We define the real-valued functions γ, p1, and p0 by

γ :=√

1 + σ − 1, p1 := γ, p0 := −3

4∂xγ, (3.16)

where σ is as in (3.1), and the main complex-valued unknown

u := |∂x|h− i|∂x|1/2ω + Tp1P≥1|∂x|h+ Tp0P≥1|∂x|−1∂xh. (3.17)

Then u satisfies the evolution equation

∂tu− i|∂x|3/2u− iΣγ(u) = −∂xTV u+N2(h, ω) + U≥3, (3.18)

where

U≥3 ∈ |∂x|1/2O3,1/2, (3.19)

the operator Σγ is given by

Σγ(u) = TγP≥1|∂x|3/2u−3

4T∂xγP≥1∂x|∂x|−1/2u, (3.20)

and the quadratic terms (expressed in h and ω) are

N2(h, ω) = −[|∂x|, ∂xT∂xω]h− iT∂2xω|∂x|1/2ω + i[|∂x|1/2, T∂xω∂x]ω

+ |∂x|M2(ω, h)− i|∂x|1/2Q2(ω, ω)− i|∂x|1/2T|∂x|3hP≥1h.(3.21)

We will express these quadratic terms as functions of u and u via (3.17) later on.

Proof. We start by calculating:

∂tu = ∂t(|∂x|h− i|∂x|1/2ω + Tp1P≥1|∂x|h+ Tp0P≥1|∂x|−1∂xh

)= |∂x|∂th− i|∂x|1/2∂tω + Tp1P≥1|∂x|∂th+ Tp0P≥1|∂x|−1∂x∂th+ |∂x|1/2O3,1/2

= |∂x|2ω − |∂x|∂xTV h+ |∂x|M2(ω, h)

+ i|∂x|5/2h+ i|∂x|3/2Tσ|∂x|h+ i|∂x|1/2TV ∂xω − i|∂x|1/2Q2(ω, ω)− i|∂x|1/2T|∂x|3hP≥1h

+ Tp1P≥1

(|∂x|2ω − |∂x|∂xTV h

)+ Tp0P≥1

(∂xω + |∂x|TV h

)+ |∂x|1/2O3,1/2.

Gathering appropriately the above terms we can write

∂tu− i|∂x|3/2u = −|∂x|∂xTV h− Tp1P≥1|∂x|∂xTV h+ Tp0P≥1|∂x|TV h+ |∂x|M2(ω, h)

+ i|∂x|1/2TV ∂xω − i|∂x|1/2Q2(ω, ω)− i|∂x|1/2T|∂x|3hP≥1h

+ Tp1P≥1|∂x|2ω + Tp0P≥1∂xω + i|∂x|3/2Tσ|∂x|h

− i|∂x|3/2Tp1P≥1|∂x|h− i|∂x|3/2Tp0P≥1|∂x|−1∂xh+ |∂x|1/2O3,1/2.

(3.22)

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18 ALEXANDRU D. IONESCU AND FABIO PUSATERI

We observe that the expression in the first two lines in the right-hand side of (3.22) is equal to

− ∂xTV u− [|∂x|, ∂xTV ]h− iT∂xV |∂x|1/2ω + i[|∂x|1/2, TV ∂x]ω − [Tp1 |∂x|P≥1, ∂xTV ]h

− [Tp0 |∂x|−1∂xP≥1, ∂xTV ]h+ |∂x|M2(ω, h)− i|∂x|1/2Q2(ω, ω)− i|∂x|1/2T|∂x|3hP≥1h

= −∂xTV u− [|∂x|, ∂xTV ]h− iT∂xV |∂x|1/2ω + i[|∂x|1/2, TV ∂x]ω

+ |∂x|M2(ω, h)− i|∂x|1/2Q2(ω, ω)− i|∂x|1/2T|∂x|3hP≥1h+ |∂x|1/2O3,1/2.

(3.23)

Using the definition of V and ω in (3.1), we see that the above quadratic terms coincide up to

|∂x|1/2O3.1/2 with the quadratic terms appearing in (3.18) with (3.21).We then look at the cubic and higher order terms in the last two lines in the right-hand side

of (3.22). Our aim is to show that they are of the form iΣγ(u), see (3.20), up to acceptableerrors. For this purpose we first use the definition of u in (3.17) and write

iΣγ(u) = iTγP≥1|∂x|5/2h+ TγP≥1|∂x|2ω + iTγP≥1|∂x|3/2Tp1P≥1|∂x|h

+ iTγP≥1|∂x|3/2Tp0P≥1|∂x|−1∂xh−3i

4T∂xγP≥1∂x|∂x|1/2h−

3

4T∂xγP≥1∂xω

− 3i

4T∂xγP≥1∂x|∂x|−1/2Tp1P≥1|∂x|h−

3i

4T∂xγP≥1∂x|∂x|−1/2Tp0P≥1|∂x|−1∂xh.

(3.24)

The last term on the last line is |∂x|1/2O3,1/2 so we can disregard it.We then compare the expression in (3.24) above and the last two lines of (3.22). Our

proposition will be proven if the identities

i|∂x|3/2Tσ|∂x|h− i|∂x|3/2Tp1P≥1|∂x|h− i|∂x|3/2Tp0P≥1|∂x|−1∂xh

= iTγP≥1|∂x|5/2h+ iTγP≥1|∂x|3/2Tp1P≥1|∂x|h+ iTγP≥1|∂x|3/2Tp0P≥1|∂x|−1∂xh

− 3i

4T∂xγP≥1∂x|∂x|1/2h−

3i

4T∂xγP≥1∂x|∂x|−1/2Tp1P≥1|∂x|h+ |∂x|1/2O3,1/2

(3.25)

and

Tp1P≥1|∂x|2ω + Tp0P≥1∂xω = TγP≥1|∂x|2ω −3

4T∂xγP≥1∂xω. (3.26)

hold true. We immediately notice that the second equation (3.26) is satisfied by imposingp1 = γ and p0 = −3∂xγ/4, as in (3.16). We then need to verify that (3.25) can be satisfied foran appropriate choice of the function γ.

We notice that all the multipliers P≥1 in (3.25) can be dropped, at the expense of acceptableerrors. Therefore (3.25) holds provided one has the following two identities for the symbols:

σ − p1 = γ + γp1, (3.27)

3

2∂xσ −

3

2∂xp1 + p0 =

3

2γ∂xp1 − γp0 +

3

4∂xγ +

3

4∂xγp1. (3.28)

Since p1 = γ, (3.27) becomes 2γ + γ2 = σ, which is satisfied by imposing the first identityin (3.16). One can then verify that the last equation (3.28) is automatically satisfied.

Remark 3.3. We notice that the symmetrization obtained in Proposition 3.2, and the formulas(3.16) for p1, p0 and γ, are simpler than the ones of [1]. This is not only because we areconsidering the 1 dimensional case, but also because of our choice of the main variables inwhich we express the system, that is the energy variables |∂x|h and |∂x|1/2ω.

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WATER WAVES WITH SURFACE TENSION 19

3.1.1. The quadratic terms. We analyze now the quadratic terms in (3.21),

N2(h, ω) =6∑

k=1

N k2

N 12 := −[|∂x|, ∂xT∂xω]h, N 2

2 := −iT∂2xω|∂x|1/2ω, N 3

2 := i[|∂x|1/2, T∂xω∂x]ω,

N 42 := −i|∂x|1/2T|∂x|3hP≥1h, N 5

2 := |∂x|M2(ω, h), N 62 := −i|∂x|1/2Q2(ω, ω).

(3.29)

Since γ =√

1 + σ − 1, using (3.17) we have

h =1

2|∂x|−1(u+ u) + P≥−4O3,1, ω = − 1

2i|∂x|−1/2(u− u). (3.30)

Using these relations we can express the quadratic terms (3.29) in terms of u and u, i.e.

N 12 =

1

4i

[|∂x|, ∂xT∂x|∂x|−1/2(u−u)

]|∂x|−1(u+ u) + |∂x|1/2O3,1/2,

N 22 =

1

4iT|∂x|3/2(u−u)(u− u),

N 32 =

1

4i[|∂x|1/2, T∂x|∂x|−1/2(u−u)∂x]|∂x|−1/2(u− u),

N 42 =

1

4i|∂x|1/2T|∂x|2(u+u)P≥1|∂x|−1(u+ u) + |∂x|1/2O3,1/2,

N 52 =

i

4|∂x|M2

(|∂x|−1/2(u− u), |∂x|−1(u+ u)

)+ |∂x|1/2O3,1/2,

N 62 =

i

4|∂x|1/2Q2(|∂x|−1/2(u− u), |∂x|−1/2(u− u)).

We divide these terms into 8 groups, by distinguishing the different types of interactions withrespect to the specific pairing of u and u, and the type of frequency interactions (Low×High→High interactions associated to the symbol χ(ξ − η, η) and High × High → Low interactionsassociated to the symbol χ(ξ − η, η)). Recall the formulas (3.13). We define

a++(ξ, η) :=1

4iχ(ξ − η, η)

[− ξ(ξ − η)

|ξ − η|1/2( |ξ||η|− 1)

+ |ξ − η|3/2 +η(ξ − η)

|ξ − η|1/2(

1− |ξ|1/2

|η|1/2)

+ |ξ − η|2 |ξ|1/2ϕ≥1(η)

|η|+|ξ|2 − |ξ|1/2|η|3/2

|η||ξ − η|1/2ϕ≤0(η)

],

(3.31)

a+−(ξ, η) :=1

4iχ(ξ − η, η)

[− ξ(ξ − η)

|ξ − η|1/2( |ξ||η|− 1)− |ξ − η|3/2 − η(ξ − η)

|ξ − η|1/2(

1− |ξ|1/2

|η|1/2)

+ |ξ − η|2 |ξ|1/2ϕ≥1(η)

|η|+|ξ|2 + |ξ|1/2|η|3/2

|η||ξ − η|1/2ϕ≤0(η)

],

(3.32)

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20 ALEXANDRU D. IONESCU AND FABIO PUSATERI

a−+(ξ, η) :=1

4iχ(ξ − η, η)

[ ξ(ξ − η)

|ξ − η|1/2( |ξ||η|− 1)− |ξ − η|3/2 − η(ξ − η)

|ξ − η|1/2(

1− |ξ|1/2

|η|1/2)

+ |ξ − η|2 |ξ|1/2ϕ≥1(η)

|η|+−|ξ|2 + |ξ|1/2|η|3/2

|η||ξ − η|1/2ϕ≤0(η)

],

(3.33)

a−−(ξ, η) :=1

4iχ(ξ − η, η)

[ ξ(ξ − η)

|ξ − η|1/2( |ξ||η|− 1)

+ |ξ − η|3/2 +η(ξ − η)

|ξ − η|1/2(

1− |ξ|1/2

|η|1/2)

+ |ξ − η|2 |ξ|1/2ϕ≥1(η)

|η|+−|ξ|2 − |ξ|1/2|η|3/2

|η||ξ − η|1/2ϕ≤0(η)

],

(3.34)

and

b++(ξ, η) =i

4

|ξ|m2(ξ, η)

|ξ − η|1/2|η|+i

4

|ξ|1/2q2(ξ, η)

|ξ − η|1/2|η|1/2, (3.35)

b+−(ξ, η) =i

4

|ξ|m2(ξ, η)

|ξ − η|1/2|η|− i

4

|ξ|1/2q2(ξ, η)

|ξ − η|1/2|η|1/2, (3.36)

b−+(ξ, η) = − i4

|ξ|m2(ξ, η)

|ξ − η|1/2|η|− i

4

|ξ|1/2q2(ξ, η)

|ξ − η|1/2|η|1/2, (3.37)

b−−(ξ, η) = − i4

|ξ|m2(ξ, η)

|ξ − η|1/2|η|+i

4

|ξ|1/2q2(ξ, η)

|ξ − η|1/2|η|1/2, (3.38)

where

m2(ξ, η) := χ(ξ − η, η)][ξ(ξ − η)− |ξ||ξ − η|

],

q2(ξ, η) := χ(ξ − η, η)η(ξ − η) + |η||ξ − η|

2.

(3.39)

Using the operator-symbol notation (2.4) and (2.16), we notice that

6∑k=1

N k2 =

∑X∈A,B

X++(u, u) +X+−(u, u) +X−+(u, u) +X−−(u, u) + |∂x|1/2O3,1/2. (3.40)

Let us also denote ∑?

:=∑

(ε1,ε2)∈(+,+),(+,−),(−,+),(−,−)

. (3.41)

For any complex-valued function f , we use the notation f+ := f , f− := f . We summarize theabove computations in the following proposition:

Proposition 3.4. Let (h, φ) be a solution of (2.24) satisfying the bootstrap assumption (2.25),and let u be defined as in (3.17) with ω, σ, V given by (3.1). Then we have

∂tu− i|∂x|3/2u− iΣγ(u) = −∂xTV u+Nu + U≥3 (3.42)

with U≥3 ∈ |∂x|1/2O3,1/2,

Σγ = TγP≥1|∂x|3/2 −3

4T∂xγP≥1∂x|∂x|−1/2, γ =

√1 + σ − 1, (3.43)

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WATER WAVES WITH SURFACE TENSION 21

and

Nu =∑?

[Aε1ε2(uε1 , uε2) +Bε1ε2(uε1 , uε2)

]. (3.44)

The symbols of the quadratic operators are given in (3.31)–(3.38). Moreover, for any t ∈ [0, T ],

〈t〉−p0K′I(t) + 〈t〉−4p0K′S(t) + 〈t〉1/2∑k∈Z

(2N2k + 2−k/10)∥∥Pku(t)

∥∥L∞. ε1, (3.45)

where, for O ∈ I, S,[K′O(t)

]2:=

∫R|Ou(ξ, t)|2 ·

(|ξ|−1 + |ξ|2NO

)P((1 + t)2|ξ|) dξ. (3.46)

The initial data u(0) satisfy the stronger bounds (recall that 0 < ε0 ≤ ε1 ≤ ε2/30 1)∑

O∈I,S

∥∥(|∂x|−1/2+p1 + |∂x|NO)Ou(0)

∥∥L2 . ε0. (3.47)

The bounds (3.45)–(3.47) follow from the apriori assumptions (2.25)–(2.27) and the definition

(3.17) (notice that u = |∂x|h− i|∂x|1/2φ at very low frequencies).We notice that the only quasilinear quadratic contributions to the nonlinearity in (3.42)

come from the term −∂xTV u on the right-hand side of (3.42). All of the other quadraticcontributions do not lose derivatives. The term −iΣγ(u) arises from the presence of surfacetension. This term loses 3/2 derivative but it is essentially a self-adjoint operator. We willexploit this structure below to perform energy estimates.

We have thus reduced the water waves system (2.24) to the equation (3.42) above for a singlecomplex valued unknown u. From now on we will work with u as our main variable. We willalso keep V as a variable and keep in mind that, in view of (3.1) and (3.30),

V = − 1

2i∂x|∂x|−1/2(u− u) + V2, V2 := ∂xTBP≥1h−Bhx. (3.48)

3.2. Higher order derivatives and weights. To implement the energy method we need tocontrol the increment of higher order Sobolev norms of the main variable u. Because of thepresence of the operator Σγ , which is of order 3/2, one cannot construct higher order energiesby applying regular derivatives to the equation. We apply instead suitably modified versionsof derivatives to the equation. The differential operator we will use, dictated by the structureof the equation, is given by D := |∂x|3/2 + Σγ , see the definition (3.43). Let k ∈ [1, 2N0/3] bean even integer and define

W = Wk := Dku, D := |∂x|3/2 + Σγ (3.49)

Below we derive the equation satisfied by W .

Proposition 3.5. Let u be the solution of (3.42)–(3.44), and let W = Wk be defined by (3.49).Let N := 3k/2. Then we have

∂tW − i|∂x|3/2W − iΣγ(W ) = QW + |∂x|NNW +OW (3.50)

where Σγ is as in (3.43), and the nonlinearities are

QW (ξ) :=1

∫RqN (ξ, η)V (ξ − η)W (η) dη, qN (ξ, η) := −iξ |ξ|

N

|η|Nχ(ξ − η, η), (3.51)

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22 ALEXANDRU D. IONESCU AND FABIO PUSATERI

and

NW =∑?

[Aε1ε2

(uε1 , |∂x|−NWε2

)+Bε1ε2

(uε1 , uε2

)]. (3.52)

The cubic nonlinearity OW satisfies the bounds

〈t〉1−p0‖OW (t)‖HN0−N,−1/2 + 〈t〉1−4p0‖SOW (t)‖HN1−N,−1/2 . ε31, if N ≤ N1,

〈t〉1−p0‖OW (t)‖HN0−N,−1/2 . ε31, if N ∈ [N1, N0].

(3.53)

Proof. The starting point is the equation (3.42). Applying Dk to this equation we see that

∂tDku− iDDku =[∂t,Dk

]u−Dk∂xTV u+DkNu +Dk|∂x|1/2O3,1/2.

Since W = Dku, we can rewrite this equation as

∂tW − iDW = −|∂x|N∂xTV |∂x|−NW + |∂x|NNW+[∂t,Dk

]u+

[∂xTV , |∂x|N

](u− |∂x|−NW ) +

[∂xTV ,Dk − |∂x|N

]u

+ (Dk − |∂x|N )Nu + |∂x|N (Nu −NW ) +Dk|∂x|1/2O3,1/2.

(3.54)

To prove the proposition it suffices to show that all the terms in the last two lines of (3.54)satisfy the cubic bounds (3.53). This is proved in Proposition C.3.

Define the weighted variable

Zk := SDku, D = |∂x|3/2 + Σγ , k ∈ [0, 2N1/3], (3.55)

where S = (3/2)t∂t + x∂x. The next lemma gives the evolution equation for the variables Zk.

Proposition 3.6. Assume k ∈ [0, 2N1/3] ∩ Z, N = 3k/2, and let Z = Zk. Then we have

∂tZ − i|∂x|3/2Z − iΣγZ = QZ +NZ,1 +NZ,2 +NZ,3 +OZ , (3.56)

where the quasilinear quadratic nonlinearity QZ is given by

QZ := QN (V,Z) = F−1[ 1

∫RqN (ξ, η)V (ξ − η)Z(η) dη

], (3.57)

and qN is as in (3.51). The quadratic semilinear terms are given by

NZ,1 := |∂x|N∑?

Aε1ε2(uε1 , |∂x|−NZε2),

NZ,2 := (i/2)QN (∂x|∂x|−1/2(Su− Su), |∂x|Nu)

+ |∂x|N∑?

[Aε1ε2(Suε1 , uε2) +Bε1ε2(Suε1 , uε2) +Bε1ε2(uε1 , Suε2)

],

NZ,3 := (i/2)QN(∂x|∂x|−1/2(u− u), |∂x|Nu

)+ (i/2)QN

(∂x|∂x|−1/2(u− u), |∂x|Nu

)+ |∂x|N (3/2−N)Nu

+ |∂x|N∑?

[NAε1ε2(uε1 , uε2) + Aε1ε2(uε1 , uε2) + Bε1ε2(uε1 , uε2)

].

(3.58)

Here we are using the definition (2.5)-(2.6) for a bilinear operator M with symbol m. Theremainder term OZ is cubic and satisfies

‖OZ(t)‖H0,−1/2 . ε31〈t〉−1+4p0 . (3.59)

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WATER WAVES WITH SURFACE TENSION 23

Proof. Using (3.50) we have

∂tWk − i|∂x|3/2Wk − iΣγWk = QN (V,Wk) + |∂x|NNu +OWk, (3.60)

with QN1 and qN1 is in (3.57), and with a remainder OWk1satisfying (3.53). Notice that

[S, ∂t − iΛ] = −(3/2)(∂t − iΛ), where Λ = |∂x|3/2. Therefore, applying S to (3.60), andcommuting it with the left-hand side, we obtain

∂tZ − i|∂x|3/2Z − iΣγZ = SQN (V,Dku) + S|∂x|NNu+ (3/2)(∂t − iΛ)Dku+ i[S,Σγ ]Dku+ SOWk

.(3.61)

Recall also the formulas (3.48) and

SQN (V,Dku) = QN (V,Z) +QN (SV,Dku) + QN (V,Dku).

Using also the commutation identities

[S, ∂x|∂x|−1/2] = −(1/2)∂x|∂x|−1/2, [S, |∂x|N ] = −N |∂x|N ,

it follows that

S|∂x|NNu = −N |∂x|NNu + |∂x|N∑?

[Aε1ε2(uε1 , Suε2) +Aε1ε2(Suε1 , uε2)

+ Aε1ε2(uε1 , uε2) +Bε1ε2(Suε1 , uε2) +Bε1ε2(uε1 , Suε2) + Bε1ε2(uε1 , uε2)].

(3.62)

Notice that some of these terms can all found in NZ,2 and NZ,3.The desired formula (3.56) follows from (3.61) provided that

OZ : = i[S,Σγ ]Dku+ SOWk+[QN (V,Dku)− i

2QN(∂x|∂x|−1/2(u− u), |∂x|Nu

)]+[3

2QN (V,Dku)− 3i

4QN(∂x|∂x|−1/2(u− u), |∂x|Nu

)]+[QN (SV,Dku)− i

2QN(∂x|∂x|−1/2(Su− Su), |∂x|Nu

)+i

4QN(∂x|∂x|−1/2(u− u), |∂x|Nu

)]+

3i

2ΣγDku+

3

2

[OWk

+ |∂x|N (NWk−Nu)

]+ |∂x|N

∑?

[Aε1ε2(uε1 , Suε2)−Aε1ε2(uε1 , |∂x|−NZε2)−NAε1ε2(uε1 , uε2)

].

(3.63)

The elliptic cubic bound (3.59) is verified in Proposition C.3.

4. Energy estimates I: high Sobolev estimates

In this section we prove the following main proposition:

Proposition 4.1. Assume that u satisfies (3.45)–(3.47). Then

supt∈[0,T ]

(1 + t)−p0‖P≥−20u(t)‖HN0 . ε0. (4.1)

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24 ALEXANDRU D. IONESCU AND FABIO PUSATERI

4.1. The higher order energy functional. Let W = Dk0u, N0 = 3k0/2, D = |∂x|3/2 + Σγ ,and recall the equations (3.42) and (3.50) for u and W ,

∂tu− i|∂x|3/2u− iΣγu = −∂xTV u+Nu + |∂x|1/2O3,1/2,

V = − 1

2i∂x|∂x|−1/2(u− u) +O2,−1/2,

∂tW − i|∂x|3/2W − iΣγW = QW + |∂x|N0NW +OW

Σγ = TγP≥1|∂x|3/2 −3

4T∂xγP≥1|∂x|−1/2∂x , γ =

√1 + σ − 1,

(4.2)

where Nu is defined in (3.44), QW is in (3.51), NW is defined in (3.52) together with (3.31)–(3.34), (3.35)–(3.38), and OW satisfies the cubic bounds (3.53). Notice that

u = |∂x|−N0W +O3,0, (4.3)

so that, using (3.45), for any t ∈ [0, T ],

‖P≥−20u(t)‖HN0 . ‖W (t)‖L2 + ε3/21 ,

‖W (t)‖L2 + ‖u‖HN0 . ε1(1 + t)p0 .(4.4)

We define the quadratic energy functional associated to the second equation in (4.2) by

E(2)N0

(t) =1

2

∫R|W (x, t)|2 dx =

1

∫RW (ξ, t)W (ξ, t) dξ. (4.5)

Based on the equation (4.2) we define the following cubic energy functional:

E(3)mN0

(t) :=1

4π2<∫R×R

W (ξ, t)W (η, t)mN0(ξ, η)u(ξ − η, t) dξdη,

mN0(ξ, η) :=(ξ − η)

[ξ|ξ|N0 |η|−N0χ(ξ − η, η)− η|η|N0 |ξ|−N0χ(η − ξ, ξ)

]2|ξ − η|1/2(|ξ|3/2 − |ξ − η|3/2 − |η|3/2)

.

(4.6)

Given the symbols aε1ε2 and bε1ε2 in (3.31)-(3.38), we also define the cubic functionals

E(3)a,ε1ε2(t) :=

1

4π2<∫R×R

W (ξ, t)Wε2(η, t)uε1(ξ − η, t) aN0ε1ε2(ξ, η) dξdη,

aN0ε1ε2(ξ, η) :=

−i|ξ|N0 |η|−N0aε1ε2(ξ, η)

|ξ|3/2 − ε2|η|3/2 − ε1|ξ − η|3/2

(4.7)

and

E(3)b,ε1ε2

(t) :=1

4π2<∫R×R

W (ξ, t)uε2(η, t)uε1(ξ − η, t) bN0ε1ε2(ξ, η) dξdη,

bN0ε1ε2(ξ, η) :=

−i|ξ|N0bε1ε2(ξ, η)

|ξ|3/2 − ε2|η|3/2 − ε1|ξ − η|3/2,

(4.8)

where, for any function f , we use the notation f+ := f , f− := f .Then the cubic correction to the energy is given by

E(3)N0

(t) := E(3)mN0

(t) +∑?

(E(3)a,ε1ε2(t) + E

(3)b,ε1ε2

(t)), (4.9)

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WATER WAVES WITH SURFACE TENSION 25

and the total energy is

EN0(t) := E(2)N0

(t) + E(3)N0

(t). (4.10)

Proposition 4.1 will follow from the two lemmas below:

Lemma 4.2. Assuming the bounds (3.45)–(3.47), for any t ∈ [0, T ] we have

|E(3)N0

(t)| . ε31(1 + t)2p0 . (4.11)

The above lemma essentially establishes the equivalence of EN0 and E(2)N0

at every fixed timeslice. The next lemma provides improved control on the increment of EN0 .

Lemma 4.3. Assuming the bounds (3.45)–(3.47), for any t ∈ [0, T ] we have

d

dtEN0(t) . ε4

1(1 + t)−1+2p0 . (4.12)

Proof of Proposition 4.1. Using (4.10) and (4.12), we see that∣∣E(2)N0

(t) + E(3)N0

(t)∣∣ ≤ ∣∣E(2)

N0(0) + E

(3)N0

(0)∣∣+

∫ t

0ε3

1(1 + s)−1+2p0 ds

for any t ∈ [0, T ]. In view of (4.11) we then have

E(2)N0

(t) . E(2)N0

(0) + ε31(1 + t)2p0 . ε3

1(1 + t)2p0 ,

for any t ∈ [0, T ], and the desired conclusion follows using also (4.4).

4.2. Analysis of the symbols and proof of Lemma 4.2. In order to prove Lemmas 4.2and 4.3, we need to establish bounds on the symbols of the cubic energy functionals in (4.6),(4.7) and (4.8). With the definition (2.3), inspecting the symbol mN0 in (4.6), using (A.13),and standard integration by parts, one can see that

‖mk,k1,k2N0

‖S∞. 2k1/22−k/21X (k, k1, k2)1[6,∞)(k2 − k1). (4.13)

Looking at the definition of the symbols in (4.7), and using the bounds (A.7) in Lemma A.1,we see that ∥∥(aN0

ε1+)k,k1,k2∥∥S∞. 2k1/22−k/21X (k, k1, k2)1[6,∞)(k2 − k1),∥∥(aN0

ε1−)k,k1,k2∥∥S∞.(23k1/22−3k/21[2,∞)(k) + 2k1/22−k/21(−∞,1](k)

)× 1X (k, k1, k2)1[6,∞)(k2 − k1),

(4.14)

while (A.18) and (4.8) give∥∥(bN0ε1ε2)k,k1,k2

∥∥S∞. 2(N0+1/2)k2−k2/21X (k, k1, k2)1[−15,15](k2 − k1). (4.15)

We now apply Lemma 2.1(ii), together with the bounds established above, to prove (4.11).Using Lemma 2.1(ii), and the notation (2.1), we can estimate the term in (4.6) as follows:∣∣E(3)

mN0(t)∣∣ . ∑

k,k1,k2∈Z‖mk,k1,k2

N0‖S∞‖P ′kW‖L2‖P ′k2W‖L2‖P ′k1u‖L∞ .

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26 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Using the bound (4.13) on mN0 , the assumptions in (3.45)–(3.47), and (4.3), we obtain∣∣E(3)mN0

(t)∣∣ . ∑

(k,k1,k2)∈X , |k−k2|≤5

2k1/22−k/2‖P ′kW‖L2‖P ′k2W‖L2ε12k1/102−k+1 〈t〉−1/2 . ε3

1〈t〉−1/3,

where x+ = max(x, 0). The cubic energies (4.7) can be dealt with in an identical fashion, sincethe bounds (4.14) on their symbols are analogous to the one for mN0 in (4.13). The cubiccorrections in (4.8) can also be treated similarly. We use Lemma 2.1(ii), (4.15), the a prioriassumptions in (3.45)–(3.47) and (4.3), to obtain, for all ε1, ε2 ∈ +,−,∣∣E(3)

bε1ε2(t)∣∣ . ∑

k,k1,k2∈Z‖bk,k1,k2ε1ε2 ‖

S∞‖P ′kW‖L2‖P ′k2W‖L2‖P ′k1u‖L∞ .

.∑

(k,k1,k2)∈X , |k1−k2|≤10

2(N0+1/2)k2−k1/2‖P ′kW‖L2‖P ′k2u‖L2ε12k1/102−k+1 〈t〉−1/2

. ε31〈t〉−1/3.

This concludes the proof of (4.11).

4.3. Proof of Lemma 4.3. Using the equation for W in (4.2) we can calculate

d

dtE

(2)N0

(t) =1

2π<∫RW (ξ, t)∂tW (ξ, t) dξ = A1(t) +A2(t) +A3(t) +A4(t),

where

A1 :=1

2π<∫RW (ξ) iΣγW (ξ) dξ, (4.16)

A2 :=1

4π2<∫R×R

W (ξ)(− iξ|ξ|N0 |η|−N0 V (ξ − η)χ(ξ − η, η)

)W (η) dξdη, (4.17)

A3 :=1

2π<∫RW (ξ) |ξ|N0NW (ξ) dξ, (4.18)

A4 :=1

2π<∫RW (ξ)OW (ξ) dξ. (4.19)

All cubic contributions coming from the above integrals are matched, up to acceptable

quartic remainder terms, with the contributions from the time evolution of E(3)N0

, see (4.9) and

(4.6)-(4.8). This fact is established through the following series of lemmas, which will alsoprove the desired estimate (4.12).

Lemma 4.4. Under the a priori assumptions (3.45)–(3.47), we have∣∣A2(t) +d

dtE(3)mN0

(t)∣∣ . ε4

1(1 + t)−1+2p0 . (4.20)

Lemma 4.5. Under the a priori assumptions (3.45)–(3.47), we have∣∣∣A3(t) +d

dt

∑?

(E(3)a,ε1ε2(t) + E

(3)b,ε1ε2

(t))∣∣∣ . ε4

1(1 + t)−1+2p0 . (4.21)

Lemma 4.6. Under the a priori assumptions (3.45)–(3.47), we have

|A1(t) +A4(t)| . ε41(1 + t)−1+2p0 . (4.22)

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WATER WAVES WITH SURFACE TENSION 27

The rest of this section is concerned with the proofs of these lemmas.

4.3.1. Proof of Lemma 4.4. We start by symmetrizing the term A2 in (4.17) using the fact thatV is real-valued,

A2 =1

8π2<∫R×R

W (ξ)W (η)V (ξ − η)

×(− iξ|ξ|N0 |η|−N0χ(ξ − η, η) + iη|η|N0 |ξ|−N0χ(η − ξ, ξ)

)dξdη.

(4.23)

Recall, see (3.48) and Definition C.1, that

V = − 1

2i∂x|∂x|−1/2(u− u) + V2, V2 = O2,−1/2. (4.24)

Thus, we can write A2 = A2,1 +A2,2 where

A2,1 :=1

4π2<∫R×R

W (ξ)W (η)u(ξ − η)qN0(ξ, η) dξdη

qN0(ξ, η) :=i(ξ − η)

2|ξ − η|1/2[ξ|ξ|N0

|η|N0χ(ξ − η, η)− η|η|N0

|ξ|N0χ(η − ξ, ξ)

],

(4.25)

and

A2,2 :=1

8π2<∫R×R

W (ξ)W (η)V2(ξ − η)a2,2(ξ, η) dξdη,

a2,2(ξ, η) := − iξ|ξ|N0

|η|N0χ(ξ − η, η) +

iη|η|N0

|ξ|N0χ(η − ξ, ξ).

(4.26)

According to (4.25), the symbol of E(3)mN0

in (4.6) is

mN0(ξ, η) =qN0(ξ, η)

i(|ξ|3/2 − |ξ − η|3/2 − |η|3/2). (4.27)

Using the equation (4.2) we can calculate

d

dtE(3)mN0

= I1 + I2 + I3 + I4

where

I1 :=1

4π2<∫R×R

W (ξ)W (η)u(ξ − η)[− i|ξ|3/2 + i|ξ − η|3/2 + i|η|3/2

]mN0(ξ, η) dξdη, (4.28)

I2 :=1

4π2<∫R×R

[iΣγW (ξ)W (η) + W (ξ)iΣγW (η)

]u(ξ − η)mN0(ξ, η) dξdη, (4.29)

I3 :=1

4π2<∫R×R

[QW (ξ)W (η) + W (ξ)QW (η)

]u(ξ − η)mN0(ξ, η) dξdη, (4.30)

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28 ALEXANDRU D. IONESCU AND FABIO PUSATERI

and

I4 :=1

4π2<∫R×R

[|ξ|N0NW (ξ)W (η) + W (ξ)|η|N0NW (η)

]u(ξ − η)mN0(ξ, η) dξdη

+1

4π2<∫R×R

W (ξ)W (η)F(∂tu− i|∂x|3/2u)(ξ − η)mN0(ξ, η) dξdη

+1

4π2<∫R×R

[OW (ξ)W (η) + W (ξ)OW (η)

]u(ξ − η)mN0(ξ, η) dξdη.

(4.31)

Using (4.25), (4.27) and (4.28) we see that A2,1 + I1 = 0, and, therefore,

A2 +d

dtE(3)mN0

= A2,2 + I2 + I3 + I4.

It then suffices to show that

|A2,2(t)|+ |I2(t)|+ |I3(t)|+ |I4(t)| . ε41(1 + t)−1+2p0 . (4.32)

Estimate of A2,2. Using integration by parts one can see that the symbol in (4.26) satisfies

a2,2(ξ, η) = O(|ξ − η|1[2−5,25](|η|/|ξ|)

),

see the notation (2.14) for bilinear symbols. Then, using Lemma 2.1(ii) and (4.24),

|A2,2| .∑

k,k1,k2∈Z‖(a2,2)k,k1,k2‖S∞‖P

′kW‖L2‖P ′k1V2‖L∞‖P

′k2W‖L2

.∑

(k,k1,k2)∈X , |k−k2|≤10

2k1‖P ′kW‖L2ε212−k

+1 〈t〉−1‖P ′k2W‖L2

. ε41〈t〉−1+2p0 .

(4.33)

Estimate of I2. The term I2 in (4.29) presents a potential loss of 3/2 derivatives. However,exploiting the structure of Σγ and of the symbol mN0 , one can recover this loss. Recall thedefinition of Σγ from (4.2). We can then estimate

|I2| .∣∣∣− ∫

R3

γ(ρ− ξ)χ(ξ − ρ, ρ)|ρ|3/2(

1 +3(ξ − ρ)

)P≥1W (ρ)W (η)u(ξ − η)mN0(ξ, η) dξdηdρ

+

∫R3

W (ξ)γ(η − ρ)χ(η − ρ, ρ)|ρ|3/2(

1 +3(η − ρ)

)P≥1W (ρ)u(ξ − η)mN0(ξ, η) dξdηdρ

∣∣∣.After changes of variables, it follows that

|I2| .∣∣∣ ∫

R3

W (ξ)W (η)u(ξ − ρ)γ(ρ− η)m′2(ξ, ρ, η) dξdηdρ∣∣∣,

m′2(ξ, ρ, η) := −χ(η − ρ, ξ)|ξ|3/2ϕ≥1(ξ)mN0(ξ + η − ρ, η)(

1 +3(η − ρ)

)+ χ(ρ− η, η)|η|3/2ϕ≥1(η)mN0(ξ, ρ)

(1 +

3(ρ− η)

).

(4.34)

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WATER WAVES WITH SURFACE TENSION 29

We then want to establish a bound for the symbol in the above expression, showing that itdoes not cause any derivatives loss. Let us write

m′2 = n1 + n2 + n3 + n4,

n1(ξ, η, ρ) :=[χ(ρ− η, η)− χ(η − ρ, ξ)

]|ξ|3/2ϕ≥1(ξ)mN0(ξ + η − ρ, η),

n2(ξ, η, ρ) := χ(ρ− η, η)[|η|3/2ϕ≥1(η)− |ξ|3/2ϕ≥1(ξ)

]mN0(ξ, ρ),

n3(ξ, η, ρ) := χ(ρ− η, η)|ξ|3/2ϕ≥1(ξ)[mN0(ξ, ρ)−mN0(ξ + η − ρ, η)

],

n4(ξ, η, ρ) :=3(ρ− η)

4ξχ(η − ρ, ξ)|ξ|3/2ϕ≥1(ξ)mN0(ξ + η − ρ, η),

n5(ξ, η, ρ) :=3(ρ− η)

4ηχ(ρ− η, η)|η|3/2ϕ≥1(η)mN0(ξ, ρ).

(4.35)

We will often use the observation

if f(ξ, η, ρ) = f1(ξ, ρ)f2(η, ρ) then ‖F−1f‖L1(R3) . ‖F−1f1‖L1(R2)‖F−1f2‖L1(R2). (4.36)

Moreover, since γ ∈ O2,0, see (C.19) and Definition C.1, we have, for any l ∈ Z,

‖Plγ‖L∞ . ε21〈t〉−12−3l+ , ‖Plγ‖L2 . ε2

1〈t〉−1/2+p02−3l+ . (4.37)

Using (4.36) and the bound (4.13) for mN0 , we see that

n1(ξ, η, ρ) = O((|ξ − ρ|3/2 + |ρ− η|3/2)1[22,∞)(|η|/|ρ− η|)1[22,∞)(|ξ|/|ξ − ρ|)

). (4.38)

Here we are using the notation (2.15), with (2.12)-(2.13). Then, using Lemma 2.1(iii) with thebound (4.38), the a priori decay assumption in (3.45) and (4.37), it is easy to show that∣∣∣ ∫

R3

W (ξ)W (η)u(ξ − ρ)γ(ρ− η)n1(ξ, η, ρ) dξdηdρ∣∣∣ . ‖W‖2L2ε

31(1 + t)−7/6. (4.39)

Moreover, the symbols n2, n4, n5 satisfy the same bound (4.38), so their contributions can alsobe estimated in the same way.

Finally, we look at n3 and we would like to prove the same symbol bound (4.38). Recall thedefinition of mN0 and qN0 from (4.27) and (4.25), and write

mN0(ξ, ρ)−mN0(ξ + η − ρ, η) = −ir1(ξ, η, ρ)− ir2(ξ, η, ρ), (4.40)

r1(ξ, η, ρ) :=qN0(ξ, ρ)− qN0(ξ + η − ρ, η)

|ξ + η − ρ|3/2 − |ξ − ρ|3/2 − |η|3/2,

r2(ξ, η, ρ) := qN0(ξ, ρ)[ 1

|ξ|3/2 − |ξ − ρ|3/2 − |ρ|3/2− 1

|ξ + η − ρ|3/2 − |ξ − ρ|3/2 − |η|3/2].

Inspecting the formula (4.25) we see that, when |η − ρ| ≤ 2−8|η|,

qN0(ξ, ρ)− qN0(ξ + η − ρ, η) = O(|ξ − ρ|5/2|ξ|−11[26,∞)(|ξ|/|ξ − ρ|)

), (4.41)

and, therefore,

r1(ξ, η, ρ) = O(|ξ − ρ|3/2|ξ|−3/21[22,∞)(|ξ|/|ξ − ρ|)

).

Moreover, one can directly verify that for |η| ≥ 26 max(|ξ − ρ|, |η − ρ|),

|ξ + η − ρ|3/2 − |ξ − ρ|3/2 − |η|3/2 − (|ξ|3/2 − |ξ − ρ|3/2 − |ρ|3/2) = O( |ξ − ρ|2 + |ρ− η|2

|η|1/2 + |ξ|1/2).

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30 ALEXANDRU D. IONESCU AND FABIO PUSATERI

It follows that

r2(ξ, η, ρ) = O( |ξ − ρ|3/2 + |ρ− η|3/2

|η|3/2 + |ξ|3/21[22,∞)(|η|/|ξ − ρ|)

),

whenever |η| ≥ 26|η− ρ|. The desired bound (4.38) follows for the symbol n3. This shows that|I2| . ε4

1〈t〉−1, and concludes the proof of the desired bound (4.32) for I2.For later use, namely to estimate I6,ε1+ in (4.64) and K1,1 in (6.21), we record below a slighly

more general result that was proved in our analysis.

Lemma 4.7. Consider the expression

I(t) =

∫R×R

[iΣγF (ξ)F (η) + F (ξ)iΣγF (η)

]u±(ξ − η)m(ξ, η) dξdη, (4.42)

with a symbol of the form

m(ξ, η) =q(ξ, η)

|ξ|3/2 ∓ |ξ − η|3/2 − |η|3/2(4.43)

which is supported on a region where 28|ξ − η| ≤ |η|. Here u+ = u, u− = u, and γ is in (4.2),(3.1). Assume that the following two properties hold:

q(ξ, η) = O(|ξ − η|3/21[26,∞)(|η|/|ξ − η|)

), (4.44)

and, whenever |η| ≥ 26|η − ρ| and |ξ| ≥ 26|ξ − ρ|,

q(ξ, ρ)− q(ξ + η − ρ, η) = O( |ξ − ρ|5/2 + |η − ρ|5/2

|ξ|+ |η|

). (4.45)

Then

|I(t)| . ‖F‖2L2ε31(1 + t)−7/6. (4.46)

Estimate of I3. Directly from the definition of I3 in (4.30) we have

|I3| .∣∣∣ ∫

R×R

[QW (ξ)W (η) + W (ξ)QW (η)

]u(ξ − η)mN0(ξ, η) dξdη

∣∣∣ (4.47)

where QW is defined in (3.51). Using (3.51) we have

|I3| .∣∣∣ ∫

R3

[ξ|ξ|N0 |ρ|−N0χ(ξ − ρ, ρ)V (ξ − ρ)W (ρ)W (η)

−W (ξ)η|η|N0 |ρ|−N0χ(η − ρ, ρ)V (η − ρ)W (ρ)]u(ξ − η)mN0(ξ, η) dξdηdρ

∣∣∣.Applying some changes of variables we get

|I3| .∣∣∣ ∫

R3

W (ξ)W (η)u(ξ − ρ)V (ρ− η)m′3(ξ, η, ρ) dξdηdρ∣∣∣,

m′3(ξ, η, ρ) := (ξ + η − ρ)|ξ + η − ρ|N0 |ξ|−N0χ(η − ρ, ξ)mN0(ξ + η − ρ, η)

− ρ|ρ|N0 |η|−N0χ(ρ− η, η)mN0(ξ, ρ).

Since V ∈ O1,−1/2, see (C.18), using Lemma 2.1(iii) and (4.13) we can estimate

|I3(t)| . I ′3(t) + ε21(1 + t)−1‖W‖2L2 (4.48)

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WATER WAVES WITH SURFACE TENSION 31

where

I ′3 =∣∣∣ ∫

R3

W (ξ)W (η)u(ξ − ρ)V (ρ− η)m′′3(ξ, η, ρ) dξdηdρ∣∣∣,

m′′3(ξ, η, ρ) := ρ[χ(η − ρ, ξ)mN0(ξ + η − ρ, η)− χ(ρ− η, η)mN0(ξ, ρ)

].

(4.49)

The main observation is that the symbol m′′3 above has a similar structure to the symbol n3 in(4.35). In particular, starting from (4.40) and using the property (4.41), it is easy to see that

m′′3(ξ, η, ρ) = O( |ξ − ρ|3/2 + |ρ− η|3/2

(|ξ|+ |η|)1/21[22,∞)(|ξ|/|ξ − ρ|)1[22,∞)(|η|/|ρ− η|)

). (4.50)

Using this bound in combination with Lemma 2.1(iii), recalling the V ∈ O1,−1/2, and using thea priori bounds, we get

|I ′3| .∑

|k1−k2|≤5, k1≥k3,k2≥k4

(2k3 + 2k4)‖P ′k1W‖L2‖P ′k2W‖L2‖P ′k3V ‖L∞‖P′k2u‖L∞

. ‖W‖2L2

∑k3,k4∈Z

(2k3 + 2k4)ε12k3/102−2k+3 〈t〉−1/2ε12k4/102−2k+4 〈t〉−1/2

. ε41‖W‖

2L2〈t〉−1.

This gives |I3| . ε41〈t〉−1+2p0 , which is the desired bound in (4.32).

We conclude this subsection with a more general lemma that follows from the same estimates.This lemma will be used later on to estimate terms like I3, which have a potential loss of onederivative, namely, I7,ε1+ in (4.59) and K1,2 in (6.22).

Lemma 4.8. Let QF be defined according to (3.51), and let

J(t) =

∫R×R

[QF (ξ)F (η) + F (ξ)QF (η)

]u±(ξ − η, t)m(ξ, η) dξdη, (4.51)

where the symbol has the form

m(ξ, η) =q(ξ, η)

|ξ|3/2 ∓ |ξ − η|3/2 − |η|3/2, (4.52)

and is supported on a region where 28|ξ − η| ≤ |η|. Assume that

q(ξ, η) = O(|ξ − η|3/21[26,∞)(|η|/|ξ − η|)

), (4.53)

and

q(ξ, ρ)− q(ξ + η − ρ, η) = O( |ξ − ρ|5/2 + |η − ρ|5/2

|ξ|+ |η|

), (4.54)

whenever |η| ≥ 26|η − ρ| and |ξ| ≥ 26|ξ − ρ|. Then

|J(t)| . ‖F‖2L2ε21(1 + t)−1. (4.55)

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32 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Estimate of I4. All of the terms in (4.31) do not lose derivatives and are not hard to estimate,given the symbol bound (4.13) on mN0 , the estimates on the nonlinear terms in Lemma C.4,and (3.53). We just show how to estimate the term

I4,1 :=

∫R×R|ξ|N0NW (ξ)W (η)u(ξ − η)mN0(ξ, η) dξdη,

the other terms being similar or easier. Applying Lemma 2.1(ii), the symbol bound (4.13), andusing the a priori estimates, and the estimate on the nonlinearity (C.36), we see that

|I4,1| .∑

k,k1,k2∈Z‖mk,k1,k2

N0‖S∞

2N0k‖P ′kNW ‖L2‖P ′k2W‖L2‖P ′k1u‖L∞

.∑

(k,k1,k2)∈Z, |k−k2|≤5

2k1/22−k/2ε212min(k,0)〈t〉−1/2+p0ε1〈t〉p0ε12k1/102−2k+1 〈t〉−1/2

. ε41〈t〉−1+2p0 .

This completes the proof of the bound (4.32) and therefore the proof of Lemma 4.4.

4.3.2. Proof of Lemma 4.5. Recall the definition of A3 in (4.18) and the definition of NW in

(3.52). Recall our definitions of the energies in E(3)a,ε1ε2 and E

(3)a,ε1ε2(t) in (4.7)-(4.8), and the

notation (3.41). Our aim is to show∣∣∣A3(t) +d

dt

∑?

(E(3)a,ε1ε2(t) + E

(3)b,ε1ε2

(t))∣∣∣ . ε4

1(1 + t)−1+2p0 . (4.56)

Calculating as in the previous section, using the evolution equations for W in (4.2), we seethat for each (ε1, ε2) ∈ (+,+), (+,−), (−,+), (−,−)

d

dtE(3)a,ε1ε2 = I5,ε1ε2 + I6,ε1ε2 + I7,ε1ε2 + I8,ε1ε2

where

I5,ε1ε2 =1

4π2<∫R×R

W (ξ)Wε2(η)uε1(ξ − η)[− i|ξ|3/2 + iε2|η|3/2 + iε1|ξ − η|3/2

]aN0ε1ε2(ξ, η) dξdη,

(4.57)

I6,ε1ε2 =1

4π2<∫R×R

[iΣγW (ξ)Wε2(η) + W (ξ) iε2ΣγWε2(η)

]uε1(ξ − η)aN0

ε1ε2(ξ, η) dξdη, (4.58)

I7,ε1ε2 =1

4π2<∫R×R

[QW (ξ)Wε2(η) + W (ξ)Qε2W (η)

]uε1(ξ − η)aN0

ε1ε2(ξ, η) dξdη, (4.59)

I8,ε1ε2 =1

4π2<∫R×R

[|ξ|N0NW (ξ)Wε2(η) + W (ξ)|η|N0N ε2

W (η)]uε1(ξ − η)aN0

ε1ε2(ξ, η) dξdη

+1

4π2<∫R×R

W (ξ)Wε2(η)F(∂tuε1 − ε1i|∂x|3/2uε1

)(ξ − η)aN0

ε1ε2(ξ, η) dξdη

+1

4π2<∫R×R

[OW (ξ)Wε2(η) + W (ξ)Oε2W (η)

]uε1(ξ − η)aN0

ε1ε2(ξ, η) dξdη.

(4.60)

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WATER WAVES WITH SURFACE TENSION 33

Here Q+W = QW , Q−W = QW , N+

W = NW , N−W = NW , O+W = OW , O−W = OW . Similarly

d

dtE

(3)b,ε1ε2

= I9,ε1ε2 + I10,ε1ε2

where

I9,ε1ε2 =1

4π2<∫R×R

W (ξ)uε2(η)uε1(ξ − η)[− i|ξ|3/2 + iε2|η|3/2 + iε1|ξ − η|3/2

]bN0ε1ε2(ξ, η) dξdη,

(4.61)

I10,ε1ε2 =1

4π2<∫R×RF(∂tW − i|∂x|3/2W

)(ξ)uε2(η)uε1(ξ − η)bN0

ε1ε2(ξ, η) dξdη

+1

4π2<∫R×R

W (ξ)F(∂tuε2 − ε2i|∂x|3/2uε2

)(η)uε1(ξ − η)bN0

ε1ε2(ξ, η) dξdη

+1

4π2<∫R×R

W (ξ)uε2(η)F(∂tuε1 − ε1i|∂x|3/2uε1

)(ξ − η)bN0

ε1ε2(ξ, η) dξdη.

(4.62)

We then see that

A3 +d

dt

∑?

(E(3)a,ε1ε2 + E

(3)b,ε1ε2

)=∑?

(I6,ε1ε2 + I7,ε1ε2 + I8,ε1ε2 + I10,ε1ε2

).

We show below how to estimate all the terms on the right-hand side above by Cε41〈t〉−1+2p0 .

Estimate of I6,ε1ε2. We start by looking at the case when ε2 = −1. We see from (4.14) that

the symbols aN0ε1− have a strong ellipticity. Then we use the bound

‖PkΣγF‖L2 . ε21〈t〉−3/423k/2‖P ′kF‖L21[−4,∞)(k), (4.63)

for any k ∈ Z, which is a consequence of (4.37) and Lemma 2.1 (ii). The potential loss of 3/2

derivatives coming from ΣγW is compensated by the smoothing property of the symbols aN0ε1−

in (4.14). As a consequence |I6,ε1−| . ε41〈t〉−7/6, as desired.

In the cases (ε1, ε2) ∈ (+,+), (−,+) we have

|I6,ε1+| .∣∣∣ ∫

R×R

[− ΣγW (ξ)W (η) + W (ξ)ΣγW (η)

]uε1(ξ − η)aN0

ε1+(ξ, η) dξdη∣∣∣. (4.64)

This is of the form (4.42)-(4.43), with F = W , m = aN0ε1+, q(ξ, η) = |ξ|N0 |η|−N0aε1+(ξ, η). We

can then apply Lemma 4.7, provided we verify its assumptions for aN0ε1+. Observe that the

symbols aε1+(ξ, η) are supported on a region where 28|ξ − η| ≤ |η|. The bound (A.5) for thesymbol aε1+(ξ, η) gives the property in (4.44). Moreover, the properties (A.9) show that thesecond assumption (4.45) is satisfied. Applying Lemma 4.7 to I6,ε1+ gives us the desired bound|I6,ε1+(t)| . ε4

1〈t〉−1+2p0 .

Estimate of I7,ε1ε2. This is similar to the estimates on I6,ε1ε2 : in the case ε2 = − we use the

simple bound ‖PkQW ‖L2 . ε1〈t〉−1/22k‖P ′kW‖L2 , which is similar to (4.63), and the gain of

3/2 derivatives in the symbols aN0ε1−. In the case ε2 = + we use Lemma 4.8 instead of Lemma

4.7. In both cases we conclude that |I7,ε1ε2(t)| . ε41(1 + t)−1+2p0 , as desired.

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34 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Estimate of I8,ε1ε2. Observe that all the terms in (4.60) do not lose derivatives, and moreoverthey are similar to the terms in (4.31). By performing the same estimates above one sees that

|I8,ε1ε2(t)| . ε41(1 + t)−1+2p0 .

Estimate of I10,ε1ε2. We notice that the symbols bNε1ε2 are smoothing and non-singular, see thedefinition (4.8) and the bound (4.15). Therefore, there are no losses of derivatives in (4.62)and all these terms are straightforward to estimate.

4.3.3. Proof of Lemma 4.6. The term A1 in (4.16) can bounded as desired because Σγ has beenconstructed as a symmetric operator up to order −1/2. To see that this is indeed the case,recall the definition of Σγ in (4.2), and write

4π2A1 = <∫RW (ξ)iP≥1W (η)γ(ξ − η)

(|η|3/2 +

3

4

(ξ − η)η

|η|1/2)χ(ξ − η, η) dηdξ

= <∫RW (ξ)W (η)γ(ξ − η) i

[1

2

(|η|3/2ϕ≥1(η)χ(ξ − η, η)− |ξ|3/2ϕ≥1(ξ)χ(η − ξ, ξ)

)+

3

4

(ξ − η)η

|η|1/2ϕ≥1(η)χ(ξ − η, η)

]dηdξ.

Notice that the symbol in the above expression is O((ξ − η)2(|ξ|+ |η|)−1/21[23,∞)(|η|/|ξ − η|)).We can then proceed, as done several times before, using Lemma 2.1(ii) and (4.37), and obtain

|A1| .∑

(k,k1,k2)∈X , |k−k2|≤5

22k12−k/2‖P ′kW‖L2‖P ′k2W‖L2‖Pk1γ‖L∞ . ε41〈t〉−1+2p0 . (4.65)

Using the cubic estimates on the term OW in (3.53), it is easy to see that also the term A4

in (4.19) satisfies the desired bound. We have then completed the proof of Lemma 4.6, andhence of Lemma 4.3. Proposition 4.1 is proved.

5. Energy estimates II: low frequencies

5.1. The basic low frequency energy. In this section we exploit the null structure of theequation to control the low frequency component of the solution u which we denote by

ulow := P≤−10u. (5.1)

With P : [0,∞)→ [0, 1] is as in (2.22), we define the energy

E(2)low(t) =

1

∫R

∣∣ulow(t, ξ)∣∣2|ξ|−1P((1 + t)2|ξ|) dξ. (5.2)

The main proposition in this section is the following:

Proposition 5.1. Assume that u satisfies (3.45)–(3.47). Then

supt∈[0,T ]

(1 + t)−2p0E(2)low(t) . ε2

0. (5.3)

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WATER WAVES WITH SURFACE TENSION 35

5.2. The cubic low frequency energy. Recall from Proposition 3.4 the equation (3.42)-(3.43), from which it follows that

∂tulow − i|∂x|3/2ulow = P≤−10(−∂xTV u+Nu) + |∂x|1/2O3,1/2. (5.4)

According to (5.4), we naturally define the cubic correction to the basic energy E(2)low as follows:

E(3)low(t) :=

∑?

1

4π2<∫R×R

u(t, ξ)|ξ|−1/2uε1(t, ξ − η)uε2(t, η)mlowε1ε2(ξ, η) dξdη, (5.5)

where the symbol is

mlowε1ε2(ξ, η) := (1 + ε1)(1 + ε2)

|ξ|1/2(ξ − η)[ξ|ξ|−1ϕt(ξ)χ(ξ − η, η)− η|η|−1ϕt(η)χ(η − ξ, ξ)

]8|ξ − η|1/2(|ξ|3/2 − |ξ − η|3/2 − |η|3/2)

− iϕt(ξ)

(aε1ε2(ξ, η) + bε1ε2(ξ, η)

)|ξ|1/2(|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2)

,

(5.6)

and we have denoted

ϕt(ξ) = ϕ≥4((1 + t)2ξ)ϕ2≤−10(ξ).

The first part of mlowε1ε2 is non-zero only for ε1 = ε2 = 1, and takes into account the nonlinear

term −∂xTV u. The symbol in the second line of (5.6) is needed to correct the nonlinear termsin Nu. Notice that no correction is needed if (1 + t)−2|ξ| ≤ 4.

The total energy for the low frequency part of the solution is given by

Elow := E(2)low + E

(3)low(t). (5.7)

Observe that under the a priori assumptions (3.45)–(3.47) we have

‖ϕ≥4((1 + t)2ξ)|ξ|−1/2u(t, ξ)‖L2 . ε1(1 + t)p0 ,

‖ϕ≤4((1 + t)2ξ)|ξ|−1/2+p1 u(t, ξ)‖L2 . ε1(1 + t)p0−2p1 .(5.8)

As in section 4, Proposition 5.1 follows from two main lemmas.

Lemma 5.2. Under the assumptions of Proposition 5.1, for any t ∈ [0, T ], we have∣∣E(3)low(t)

∣∣ . ε31(1 + t)2p0 . (5.9)

Lemma 5.3. Under the assumptions of Proposition 5.1, for any t ∈ [0, T ], we have∣∣∣ ddtElow(t)

∣∣∣ ≤ Cε31(1 + t)−1+2p0 + 40p1E

(2)low(t)(1 + t)−1. (5.10)

5.3. Analysis of the symbols and proof of Lemma 5.2. We first show that the symbolin (5.6) satisfies the bounds

‖(mlowε1ε2)k,k1,k2‖

S∞. 2−max(k1,k2)/21X (k, k1, k2)1[−2 log2(2+t)−10,0](k) (5.11)

and

‖mlowε1ε2ϕk1(ξ − η)ϕk2(η)‖

S∞. 2−max(k1,k2)/2, (5.12)

for all k, k1, k2 ∈ Z.

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36 ALEXANDRU D. IONESCU AND FABIO PUSATERI

We start with the first component of mlowε1ε2 , which we denote by

n1(ξ, η) :=|ξ|1/2(ξ − η)

[ξ|ξ|−1ϕt(ξ)χ(ξ − η, η)− η|η|−1ϕt(η)χ(η − ξ, ξ)

]2|ξ − η|1/2(|ξ|3/2 − |ξ − η|3/2 − |η|3/2)

.

Since χ(ξ − η, η) forces 23|ξ − η| ≤ |η|, through Taylor expansions and standard integration byparts, we see that

ξ|ξ|−1ϕt(ξ)χ(ξ − η, η)− η|η|−1ϕt(η)χ(η − ξ, ξ)|ξ|3/2 − |ξ − η|3/2 − |η|3/2

= O(|η|−3/21[22,∞](|η|/|ξ − η|)1[(1+t)−2,0](|ξ|)

),

and deduce

n1(ξ, η) = O(|ξ − η|1/2|η|−11[22,∞](|η|/|ξ − η|)1[(1+t)−2,0](|ξ|)

).

Let

n2(ξ, η) := −i ϕt(ξ)aε1ε2(ξ, η)

|ξ|1/2(|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2).

Using the bound (A.7) in Lemma A.1, one can see that

n2(ξ, η) = O(|ξ − η|1/2|ξ|−11[22,∞)(|η|/|ξ − η|)1[(1+t)−2,0](|ξ|)

),

which is a better bound than what we need. Finally, let

n3(ξ, η) := −i ϕt(ξ)bε1ε2(ξ, η)

|ξ|1/2(|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2),

where the symbols bε1ε2 are defined in (3.35)–(3.39). Using the bound (A.17) in Lemma A.2,we know that on the support of bε1ε2 the sizes of |η| and |ξ − η| are comparable, so

n3(ξ, η) = O(|ξ − η|−1/21[2−15,215](|η|/|ξ − η|)1[(1+t)−2,0](|ξ|)

). (5.13)

This suffices to obtain (5.11). Similar arguments, reexamining the formulas (3.35)-(3.38), andintegration by parts, also give (5.12) (only the bound for n3 requires an additional argument).

We now use (5.11) to prove (5.9). Let us denote

k− := min(k1, k2), k+ := max(k1, k2). (5.14)

Using the formula (5.5), the bound (5.11) together with Lemma (2.1), we estimate∣∣E(3)low(t)

∣∣ .∑?

∑(1+t)−2≤2k+10≤210

‖(mlowε1ε2)k,k1,k2‖

S∞2−k/2‖P ′ku‖L2‖P ′k+u‖L2‖P ′k−u‖L∞ .

Using the a priori assumptions (3.45)–(3.47) (see also (5.8)) we obtain∣∣E(3)low(t)

∣∣ . ∑(k,k1,k2)∈X , (1+t)−2≤2k+10

2−k/2‖P ′ku‖L22−k+/2‖P ′k+u‖L2

× ε12k−/102−N2 max(k−,0)〈t〉−1/2 . ε31〈t〉−1/3.

(5.15)

This completes the proof of Lemma 5.2.

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WATER WAVES WITH SURFACE TENSION 37

5.4. Proof of Lemma 5.3. Using the definitions (5.2),(2.22), (5.5), the equation (5.4), anda symmetrization argument similar to the one performed for the term (4.17) and leading to(4.25), we calculate

d

dt

(E

(2)low + E

(3)low

)=

1

4πJ1 + <

[ 1

2πJ2 +

1

4π2

∑?

(J3,ε1ε2 + J4,ε1ε2 + J5,ε1ε2 + J6,ε1ε2

)]+R,

J1 =

∫R

∣∣ulow(ξ)∣∣2 P ′((1 + t)2|ξ|) 2(1 + t) dξ,

J2 =

∫Rulow(ξ)F

(∂tulow − i|∂x|3/2ulow

)(ξ)ϕ≤3((1 + t)2ξ)|ξ|−1P((1 + t)2|ξ|) dξ,

J3,ε1ε2 =

∫R×R|ξ|−1/2u(ξ) ∂tm

lowε1ε2(ξ, η)uε1(ξ − η)uε2(η) dξdη,

J4,ε1ε2 =

∫R×R|ξ|−1/2F

(∂tu− i|∂x|3/2u

)(ξ)mlow

ε1ε2(ξ, η)uε1(ξ − η)uε2(η) dξdη,

J5,ε1ε2 =

∫R×R|ξ|−1/2u(ξ)mlow

ε1ε2(ξ, η)F(∂tuε1 − iε1|∂x|3/2uε1

)(ξ − η)uε2(η) dξdη,

J6,ε1ε2 =

∫R×R|ξ|−1/2u(ξ)mlow

ε1ε2(ξ, η)uε1(ξ − η)F(∂tuε2 − iε2|∂x|3/2uε2

)(η) dξdη.

(5.16)

Here R denotes a quartic term which includes the contribution from |∂x|1/2O3,1/2 in (5.4),and from the quadratic part of V , that is V2 in (4.24), and can be easily seen to satisfy

|R(t)| . ε41(1 + t)−1+2p0 . The term J1 comes from differentiating the cutoff P in the quadratic

energy, whereas J2 is obtained when we differentiate |ulow|2 in the region |ξ| . (1 + t)−2.The term J3,ε1ε2 comes from differentiating the symbol in the cubic energy functional. Theremaining three terms in (5.16) are the net result of differentiating the quadratic energy when

|ξ| & (1 + t)−2, and its cubic correction, after symmetrizations and cancellations.

5.4.1. Estimate of J1. Using (2.22), the definition of E(2)low in (5.2), and the a priori assumption

(3.45)–(3.47), we immediately see that

1

4π|J1| ≤

1

∫R

∣∣ulow(t, ξ)∣∣2 20p1

(1 + t)|ξ|P((1 + t)2|ξ|) dξ ≤ 20p1

1 + tE

(2)low(t),

as desired.

5.4.2. Estimate of J2. Notice that the term J2 in (5.16) is supported on a region where |ξ| .(1 + t)−2. Then we use the bounds in Lemma C.4, and the second bound in (5.8), to see that

|J2| .∑

2k−6≤〈t〉−2

2(−1+2p1)k‖P ′ku(t)‖L2‖P ′k(∂tu− i|∂x|3/2u

)(t)‖

L2〈t〉4p1

. 〈t〉4p1∑

2k.〈t〉−2

2(−1+2p1)k‖P ′ku(t)‖L2 ε212k/2〈t〉−1+p0

. 〈t〉−1+p0+2p1ε21

∑2k.〈t〉−2

2p1k〈t〉2p12−k/2+p1k‖P ′ku(t)‖L2

. ε31〈t〉−1+2p0 .

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38 ALEXANDRU D. IONESCU AND FABIO PUSATERI

5.4.3. Estimate of J3,ε1ε2. Recall the notation (5.14). Using the definition of mlowε1ε2 in (5.6),

and the same arguments in the proof of Lemma 5.2 that gave us (5.11), one can see that

‖(∂tmε1ε2)k,k1,k2‖S∞ . 1X (k, k1, k2)(1 + t)−12−max(k1,k2)/21(−2 log2(2+t)−10,0](k).

This is better by a factor of (1 + t)−1 than the corresponding bound on mε1ε2 . The same

argument as in the proof of Lemma 5.2 (see (5.15)) shows that |J3,ε1ε2 | . ε31〈t〉−4/3, as desired.

5.4.4. Estimate of J4,ε1ε2. We use Lemma 2.1 to estimate, for every ε1, ε2 = ±,

|J4,ε1ε2 | .∑

k,k1,k2∈Z‖(mlow

ε1ε2)k,k1,k2‖S∞

2−k/2‖P ′k(∂t − i|∂x|3/2)u‖L2‖P ′k+u‖L2‖P ′k−u‖L∞ .

Using the bound (5.11), Lemma C.4, and (3.45), we have

|J4,ε1ε2 | . ε31〈t〉−1+p0

∑(k,k1,k2)∈X , 〈t〉−2≤2k≤1

2−k+/2‖P ′k+u‖L22k−/102−N2 max(k−,0) . ε41〈t〉−1+2p0 .

5.4.5. Estimates of J5,ε1ε2 and J6,ε1ε2. Since the terms J5,ε1ε2 and J6,ε1ε2 are similar, we onlyestimate the first one. We begin again by using Lemma 2.1, and estimate, for every ε1, ε2,

|J5,ε1ε2 | . ‖ϕ≥0((1 + t)2ξ)|ξ|−1/2u‖L2

∑k1,k2

‖mlowε1ε2ϕk1(ξ − η)ϕk2(η)‖

S∞

×‖P ′k1(∂t − iΛ)u‖L∞‖P ′k2u‖L2 .

We then use the bound (5.12), Lemma C.4, and the a priori bounds (3.45)–(3.47), to obtain

|J5,ε1ε2 | . ε41〈t〉−1+2p0

∑k1,k2

2−max(k1,k2)/22k1/22−max(k1,0)2(1/2−p0)k22−max(k2,0) . ε41〈t〉−1+2p0 .

The proof of Lemma 5.3 is completed, and Proposition 5.1 follows.

6. Energy estimates III: weighted estimates for high frequencies

In this section we want to control the high frequency component of the weighted Sobolevnorm of our solution u. The main result of this section is the following weighted energy estimate:

Proposition 6.1. Assume that u satisfies (3.45)-(3.47). Then

supt∈[0,T ]

(1 + t)−4p0‖P≥−20Su(t)‖HN1 . ε0. (6.1)

6.1. The weighted energy functionals. Let Z = SDk1u, N1 = 3k1/2, D = |∂x|3/2 + Σγ .The quadratic energy we associate to the equation (3.56) is

E(2)Z (t) :=

1

∫RZ(t, ξ)Z(t, ξ)ϕ2

≥−20(ξ) dξ. (6.2)

The natural cubic energy is constructed as the sum of three energy functionals,

E(3)Z (t) = EZ,1(t) + EZ,2(t) + EZ,3(t). (6.3)

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WATER WAVES WITH SURFACE TENSION 39

The first energy functional is is the natural correction associated to the nonlinearities QZ andNZ,1 in (3.57)-(3.58),

EZ,1(t) :=1

4π2

∑?

<∫R×R

Z(ξ, t)Zε2(η, t)m1,ε1ε2(ξ, η)uε1(ξ − η, t) dξdη,

m1,ε1ε2(ξ, η) := −i|ξ|N1 |η|−N1ϕ2

≥−20(ξ)aε1ε2(ξ, η)

|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2+ (1 + ε1)(1 + ε2)

×(ξ − η)

[ξ|ξ|N1 |η|−N1ϕ2

≥−20(ξ)χ(ξ − η, η)− η|η|N1 |ξ|−N1ϕ2≥−20(η)χ(η − ξ, ξ)

]8|ξ − η|1/2(|ξ|3/2 − |ξ − η|3/2 − |η|3/2)

,

(6.4)

where the second part of the symbol, which is only present for ε1 = ε2 = 1, is associated toQZ , while the first one corresponds to NZ,1. The second functional takes into account NZ,2 in(3.58) and is defined as

EZ,2(t) :=1

4π2

∑?

<∫R×R

Z(ξ, t)uε2(η, t)m2,ε1ε2(ξ, η)Suε1(ξ − η, t) dξdη,

m2,ε1ε2(ξ, η) :=ε1(1 + ε2)(ξ − η)ξ|ξ|N1ϕ2

≥−20(ξ)χ(ξ − η, η)

4|ξ − η|1/2(|ξ|3/2 − ε1|ξ − η|3/2 − |η|3/2)

+−i|ξ|N1ϕ2

≥−20(ξ)(aε1ε2(ξ, η) + bε1ε2(ξ, η) + bε1ε2(ξ, ξ − η)

)|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2

.

(6.5)

The last functional is

EZ,3(t) :=1

4π2

∑?

<∫R×R

Z(ξ, t)uε2(η, t)m3,ε1ε2(ξ, η)uε1(ξ − η, t) dξdη,

m3,ε1ε2(ξ, η) :=iε1(1 + ε2)(ξ − η)ϕ2

≥−20(ξ)(qN1(ξ, η)|η|N1 + qN1(ξ, η)|η|N1

)4|ξ − η|1/2(|ξ|3/2 − ε1|ξ − η|3/2 − |η|3/2)

+−i|ξ|N1ϕ2

≥−20(ξ)(3/2−N1)(aε1ε2(ξ, η) + bε1ε2(ξ, η)

)|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2

+−i|ξ|N1ϕ2

≥−20(ξ)(N1aε1ε2(ξ, η) + aε1ε2(ξ, η) + bε1ε2(ξ, η)

)|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2

,

(6.6)

and is associated to the nonlinear term NZ,3 in (3.58). The total weighted energy is

EZ(t) := E(2)Z (t) + E

(3)Z (t). (6.7)

Using (3.45), the definitions, and Propositions 4.1 and 5.1, we have

‖P≥−20Su(t)‖HN1 . ‖P≥−20Z(t)‖L2 + ε3/21 (1 + t)p0 ,

‖Su(t)‖HN1 + ‖Z(t)‖L2 . ε1(1 + t)4p0 .(6.8)

Therefore, Proposition 6.1 follows from the following two main lemmas:

Lemma 6.2. Under the assumptions of Proposition 6.1, for any t ∈ [0, T ] we have

|E(3)Z (t)| . ε3

1(1 + t)8p0 . (6.9)

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40 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Lemma 6.3. Under the assumptions of Proposition 6.1, for any t ∈ [0, T ] we have

d

dtEZ(t) . ε4

1(1 + t)−1+8p0 . (6.10)

6.2. Analysis of the symbols and proof of Lemma 6.2. To prove Lemmas 6.2 and 6.3,we need bounds for the symbols of the cubic energy functionals in (6.4)-(6.6). The first termin the symbol m1,ε1ε2 in (6.4) can be estimated by using directly (A.7), while the second partis similar to the symbol mN0 in (4.6). It follows that

‖mk,k1,k21,ε1+ ‖S∞ . 2k1/22−k/21X (k, k1, k2)1[6,∞)(k2 − k1),

‖mk,k1,k21,ε1− ‖S∞ .

(23k1/22−3k/2 + 2k1/22−k/21(−∞,1](k)

)1X (k, k1, k2)1[6,∞)(k2 − k1).

(6.11)

Using the bounds (A.7) and (A.18) it is easy to see that

‖mk,k1,k22,ε1ε2

‖S∞. 1X (k, k1, k2)2N1k2k/2−k1/21[−20,∞)(k2 − k). (6.12)

Moreover, for any k, k2 ∈ Z,

‖|ξ − η|1/2m2,ε1ε2(ξ, η)ϕk(ξ)ϕk2(η)‖S∞ . 2(N1+1/2)k1[−20,∞)(k2 − k). (6.13)

Finally, for the symbols m3,ε1ε2 , we have the similar bounds

‖mk,k1,k23,ε1ε2

‖S∞. 1X (k, k1, k2)2N1k2k/2−k1/21[−20,∞)(k2 − k), (6.14)

and

‖|ξ − η|1/2m3,ε1ε2ϕk(ξ)ϕk2(η)‖S∞ . 2(N1+1/2)k1[−20,∞)(k2 − k). (6.15)

The proof of (6.9) is similar to the proof of (4.11) in Lemma 4.2 and (5.9) in Lemma 5.2,using the bounds (6.11)-(6.14) on the symbols, and Lemma 2.1(ii).

6.3. Proof of Lemma 6.3. We start by computing the time evolution of EZ in (6.7). Using

the definitions of the energies E(2)Z in (6.2), and E

(3)Z in (6.3)-(6.6), and the evolution equation

for Z derived in Lemma 3.6, see (3.56)-(3.59), we can calculate:

d

dt

(E

(2)Z + E

(3)Z

)= K0 +

1

4π2<∑?

(K1,ε1ε2 +K2,ε1ε2 +K3,ε1ε2

)+R, (6.16)

where

K0 :=1

2π<∫RZ(ξ) iΣγZ(ξ)ϕ2

≥−20(ξ) dξ, (6.17)

K1,ε1ε2 :=

∫R×RF(∂tZ − iΛZ

)(ξ)m1,ε1ε2(ξ, η) uε1(ξ − η)Zε2(η) dξdη

+

∫R×R

Z(ξ)m1,ε1ε2(ξ, η)F(∂tuε1 − iε1Λuε1

)(ξ − η)Zε2(η) dξdη

+

∫R×R

Z(ξ)m1,ε1ε2(ξ, η) uε1(ξ − η)F(∂tZε2 − iε2ΛZε2

)(η) dξdη,

(6.18)

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WATER WAVES WITH SURFACE TENSION 41

with m1,ε1ε2 as in (6.4),

K2,ε1ε2 = K12,ε1ε2 +K2

2,ε1ε2 +K32,ε1ε2 ,

K12,ε1ε2 :=

∫R×RF(∂tZ − iΛZ

)(ξ)m2,ε1ε2(ξ, η) Suε1(ξ − η)uε2(η) dξdη,

K22,ε1ε2 :=

∫R×R

Z(ξ)m2,ε1ε2(ξ, η)F(∂tSuε1 − iε1ΛSuε1

)(ξ − η)uε2(η) dξdη,

K32,ε1ε2 :=

∫R×R

Z(ξ)m2,ε1ε2(ξ, η) Suε1(ξ − η)F(∂tuε2 − iε2Λuε2

)(η) dξdη,

(6.19)

where m2,ε1ε2 is defined in (6.5), and

K3,ε1ε2 = K13,ε1ε2 +K2

3,ε1ε2 +K33,ε1ε2 ,

K13,ε1ε2 :=

∫R×RF(∂tZ − iΛZ

)(ξ)m3,ε1ε2(ξ, η) uε1(ξ − η)uε2(η) dξdη

K23,ε1ε2 :=

∫R×R

Z(ξ)m3,ε1ε2(ξ, η)F(∂tuε1 − iε1Λuε1

)(ξ − η)uε2(η) dξdη

K23,ε1ε2 :=

∫R×R

Z(ξ)m3,ε1ε2(ξ, η) uε1(ξ − η)F(∂tuε2 − iε2Λuε2

)(η) dξdη,

(6.20)

where m3,ε1ε2 is defined in (6.6). The remainder R comes from quartic terms involving theremainder OZ in (3.56), and quartic terms involving the quadratic part of the function V as itappears in (3.57), that is V2 in (4.24). Using the estimate (3.59) for the first, and argumentssimilar to the one used for A2,2, see (4.26) and (4.33), for the second, we see that

|R(t)| . ε41(1 + t)−1+8p0 .

We now show that all of the terms in (6.18)-(6.20) are bounded by ε41(1 + t)−1+8p0 as well.

6.3.1. Estimate of K0. The term in (6.17) is like to A1 in (4.16), with Z instead of W . We can

then estimate it in the same way we estimatedA1 in subsection 4.3.3, |K0(t)| . ‖Z(t)‖2L2ε21〈t〉−1.

6.3.2. Estimate of K1,ε1ε2. We rearrange the integrals in (6.18), using the equation (3.56) forZ, and identify the terms that could potentially lose derivatives and need additional arguments.Let NZ := NZ,1 +NZ,2 +NZ,3, see (3.58), and write∑

?

K1,ε1ε2 = K1,1 +K1,2 +K1,3

where

K1,1 :=∑ε1=±

∫R×R

[iΣγZ(ξ)Z(η) + Z(ξ)iΣγZ(η)

]m1,ε1+(ξ, η)uε1(ξ − η) dξdη, (6.21)

K1,2 :=∑ε1=±

∫R×R

[QZ(ξ)Z(η) + Z(ξ)QZ(η)

]m1,ε1+(ξ, η)uε1(ξ − η) dξdη, (6.22)

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42 ALEXANDRU D. IONESCU AND FABIO PUSATERI

and

K1,3 :=∑ε1=±

∫R×R

[NZ(ξ)Z(η) + Z(ξ)NZ(η)

]m1,ε1+(ξ, η)uε1(ξ − η) dξdη

+∑ε1=±

∫R×R

[OZ(ξ)Z(η) + Z(ξ)OZ(η)

]m1,ε1+(ξ, η)uε1(ξ − η) dξdη,

(6.23)

K1,4 :=∑?

∫R×R

Z(ξ)m1,ε1ε2(ξ, η)F(∂tuε1 − iε1Λuε1

)(ξ − η)Zε2(η) dξdη

+∑ε1=±

∫R×RF(∂tZ − iΛZ

)(ξ)m1,ε1−(ξ, η) uε1(ξ − η)Z−(η) dξdη

+∑ε1=±

∫R×R

Z(ξ)m1,ε1−(ξ, η) uε1(ξ − η)F(∂tZ− + iΛZ−

)(η) dξdη.

(6.24)

Estimates of K1,1 and K1,2. These terms are estimated using Lemmas 4.7 and 4.8 with F = Z,

|K1,1|+ |K1,2| . ε41(1 + t)−1+8p0

The symbol conditions on the multipliers m1,ε1+ are satisfied because the two components

of m1,ε1+ are similar to the multipliers aN0ε1+ and mN0 defined in (4.6)-(4.7), and the desired

properties have been verified in section 4.

Estimates of K1,3 and K1,4. In all the terms appearing in (6.23) and (6.24) there are no lossesof derivatives since NZ is a semilinear term, and the symbols m3,ε1− are strongly elliptic, as wecan see from the second bound in (6.11). The desired estimate follows by the same argumentsas before, using Lemma 2.1 (ii). We always estimate Z, NZ , OZ , and ∂tZ − iΛZ in L2 (using(6.8), (C.40), (3.59), and (C.39)) and u and ∂tu− iΛu in L∞ (using (3.45) and (C.35)).

6.3.3. Estimate of K2,ε1ε2. The main difficulty in estimating the terms in (6.19) comes from thesingularity in the symbol m2,ε1ε2 , see (6.12). We can overcome this using the low frequenciesinformation in (3.45). We begin with the first term in (6.19), and split it into two piecesdepending on the size of the frequency ξ − η,

K12,ε1ε2 = I + II,

I :=

∫R×RF(∂tZ − iΛZ

)(ξ)m2,ε1ε2(ξ, η)ϕ≤0((1 + t)2(ξ − η))Suε1(ξ − η)uε2(η) dξdη,

II :=

∫R×RF(∂tZ − iΛZ

)(ξ)m2,ε1ε2(ξ, η)ϕ≥1((1 + t)2(ξ − η))Suε1(ξ − η)uε2(η) dξdη.

We observe that, in view of (3.45), we have

‖PlSu‖L∞ . 2l(1−p1)ε1(1 + t)4p0−2p1 , for 2l . (1 + t)−2.

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WATER WAVES WITH SURFACE TENSION 43

Then, to estimate the first term it suffices to use Lemma 2.1(ii) together with the bound onm2,ε1ε2 in (6.12), followed by (C.39), and the a priori assumptions (3.45):

|I| .∑

k,k1,k2∈Z, 2k1.〈t〉−2

‖mk,k1,k22,ε1ε2

‖S∞‖P ′k(∂t − iΛ)Z‖L2‖P ′k1Su‖L∞‖P

′k2u‖L2

.∑

(k,k1,k2)∈X , 2k1.〈t〉−2, k2≥k−20

2k/22−k1/22N1kε21〈t〉−1/2+4p02k/22max(k,0)

× ε12k1(1−p1)〈t〉4p0−2p1ε12−N0k+2 〈t〉p0 . ε4

1〈t〉−1.

To estimate II we use Lemma 2.1(ii) together with the symbol bound in (6.13), (C.39), the apriori assumption (3.45), and the inequality N2 ≥ N1 + 2:

|II| . ‖P≥−2 log2(1+t)−5|∂x|−1/2Su‖L2

∑k,k2∈Z

‖|ξ − η|1/2m2,ε1ε2(ξ, η)ϕk(ξ)ϕk2(η)‖S∞

×‖P ′k(∂t − iΛ)Z‖L2‖P ′k2u‖L∞. ε1〈t〉4p0

∑k,k2∈Z, k2≥k−20

2(N1+1/2)kε21〈t〉−1/2+4p02k/22k

+‖P ′k2u‖L∞

. ε41(1 + t)−1+8p0 .

The terms K22,ε1ε2

and K32,ε1ε2

in (6.19) are similar (in fact easier). We provide the details

only for K32,ε1ε2

. We first write

K32,ε1ε2 = III + IV,

III :=

∫R×R

Z(ξ)m2,ε1ε2(ξ, η)ϕ≤0((1 + t)2(ξ − η))Suε1(ξ − η)F(∂tuε2 − iε2Λuε2

)(η) dξdη,

IV :=

∫R×R

Z(ξ)m2,ε1ε2(ξ, η)ϕ≥1((1 + t)2(ξ − η))Suε1(ξ − η)F(∂tuε2 − iε2Λuε2

)(η) dξdη.

Then we use Lemma 2.1(ii), the symbol bound (6.12), the estimates (C.35), and the a prioriassumptions (3.45) to obtain

|III| .∑

k,k1,k2∈Z, 2k1≤(1+t)−2

‖mk,k1,k22,ε1ε2

‖S∞‖P ′kZ‖L2‖P ′k1Su‖L∞‖P

′k2(∂t − iΛ)u‖

L2

. ε41〈t〉−1/2+9p0

∑(k,k1,k2)∈X , 2k1≤〈t〉−2, k2≥k−20

2N1k2k/22−k1/2 · 2k1(1−p1)2k2/22−(N0−2)k+2

. ε41〈t〉−1.

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44 ALEXANDRU D. IONESCU AND FABIO PUSATERI

To estimate IV we use again Lemma 2.1(ii), this time together with the bound (6.13), the lowfrequency assumption in (3.45), and the estimates (C.35), and see that

|IV | . ‖P≥−2 log2(1+t)−5|∂x|−1/2Su‖L2

∑k,k2∈Z

‖|ξ − η|1/2m2,ε1ε2(ξ, η)ϕk(ξ)ϕk2(η)‖S∞

×‖P ′kZ‖L2‖(∂t − iΛ)P ′k2u‖L∞. ε1〈t〉4p0

∑k,k2∈Z, k2≥k−20

2(N1+1/2)kε1〈t〉4p0‖(∂t − iΛ)P ′k2u‖L∞

. ε41〈t〉−1+8p0 .

This concludes the desired estimate for the integrals in (6.19).

6.3.4. Estimate of K3,ε1ε2. We observe that the symbols m3,ε1ε2 in (6.20) satisfy the same

bounds as the symbols m2,ε1ε2 , see (6.12)-(6.13) and (6.14)-(6.15). Moreover, the terms Kj3,ε1ε2

,

j = 1, 2, 3, in (6.20) are trilinear expressions of (Z, u, u), while the terms Kj2,ε1ε2

, j = 1, 2, 3,

in (6.19) are trilinear expressions of (Z, u, Su). Thus, it is clear that estimating the terms in(6.20) is easier than estimating those in (6.19), because the bounds we assume on u are strongerthan those we have on Su. This concludes the proof of Proposition 6.1.

7. Energy estimates IV: weighted estimates for low frequencies

In this section we improve our control on the low frequency component of Su. The basicquadratic energy is

E(2)Sulow

(t) =1

∫R

∣∣Su(t, ξ)∣∣2|ξ|−1P((1 + t)2|ξ|)ϕ2

≤−10(ξ) dξ, (7.1)

where P is defined in (2.22). We will prove the following:

Proposition 7.1. Assume that u satisfies (3.45)–(3.47). Then

supt∈[0,T ]

(1 + t)−8p0E(2)Sulow

(t) . ε20. (7.2)

7.1. The cubic low frequency weighted energy. We start from Proposition 3.6 with k = 0and apply the projection operator P≤−10. It follows that

P≤−10(∂t − iΛ)(Su) = P≤−10Q0(V, Su) + P≤−10NSu,1 + P≤−10NSu,2 + P≤−10OSu. (7.3)

Here the symbol of Q0 is q0(ξ, η) = −iξχ(ξ − η, η), the nonlinear terms are

NSu,1 := (i/2)Q0

(∂x|∂x|−1/2(Su− Su), u

)+∑?

[Aε1ε2(Suε1 , uε2) +Aε1ε2(uε1 , Suε2) +Bε1ε2(Suε1 , uε2) +Bε1ε2(uε1 , Suε2)

],

NSu,2 := (i/2)Q0(∂x|∂x|−1/2(u− u, u)) + (i/2)Q0

(∂x|∂x|−1/2(u− u), u

)+∑?

[Aε1ε2(uε1 , uε2) + (3/2)Aε1ε2(uε1 , uε2) + Bε1ε2(uε1 , uε2) + (3/2)Bε1ε2(uε1 , uε2)

],

(7.4)

and the remainder satisfies

‖|∂x|−1/2OSu(t)‖L2 . ε31(1 + t)−1+4p0 . (7.5)

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WATER WAVES WITH SURFACE TENSION 45

The natural cubic energy associated to (7.3)-(7.4) is given by

E(3)Sulow

= ESulow,1 + ESulow,2, (7.6)

where, with ϕt(ξ) = ϕ≥4((1 + t)2ξ)ϕ2≤−10(ξ),

ESulow,1(t) :=1

4π2

∑?

<∫R×R|ξ|−1/2Su(ξ, t)Suε2(η, t)uε1(ξ − η, t)qε1ε2(ξ, η) dξdη,

qε1ε2(ξ, η) := (1 + ε1)(1 + ε2)|ξ|1/2(ξ − η)

[ξ|ξ|−1ϕt(ξ)χ(ξ − η, η)− η|η|−1ϕt(η)χ(η − ξ, ξ)

]8|ξ − η|1/2(|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2)

+i(1 + ε1)ε2ϕt(ξ) η q0(ξ, ξ − η)

4|η|1/2|ξ|1/2(|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2)

− iϕt(ξ)

(aε1ε2(ξ, ξ − η) + aε1ε2(ξ, η) + bε1ε2(ξ, ξ − η) + bε1ε2(ξ, η)

)|ξ|1/2(|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2)

,

(7.7)

and

ESulow,2(t) :=1

4π2

∑?

<∫R×R|ξ|−1/2Su(ξ, t)uε2(η, t)rε1ε2(ξ, η)uε1(ξ − η, t) dξdη,

rε1ε2(ξ, η) :=iε1(1 + ε2)ϕt(ξ)(ξ − η)[q0(ξ, η) + q0(ξ, η)]

4|ξ − η|1/2|ξ|1/2(|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2)

+−iϕt(ξ)

(aε1ε2(ξ, η) + (3/2)aε1ε2(ξ, η) + bε1ε2(ξ, η) + (3/2)bε1ε2(ξ, η)

)|ξ|1/2(|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2)

,

(7.8)

where we are using the notation (2.6). The first part of the symbol qε1ε2 in (7.7) is needed tocorrect the quadratic interaction Q0(V, Su) in (7.3), after the proper symmetrization. This issimilar to the symbols in (4.6), (5.6), and (6.4). The rest of the symbol qε1ε2 takes into accountthe nonlinear term NSulow,1 in (7.4). The symbol rε1ε2 is naturally associated to the nonlinearterm NSulow,2 in (7.4).

The estimate (7.2) follows from the two lemmas below.

Lemma 7.2. Under the assumptions of Proposition 7.1, for any t ∈ [0, T ] we have

|E(3)Sulow

(t)| . ε31(1 + t)8p0 . (7.9)

Lemma 7.3. Under the assumptions of Proposition 7.1, for any t ∈ [0, T ] we have

d

dtESulow(t) ≤ Cε3

1(1 + t)−1+8p0 + 40p1E(2)Sulow

(t)(1 + t)−1. (7.10)

7.2. Analysis of the symbols and proof of Lemma 7.2. To prove Lemma 7.2 and 7.3 weneed appropriate bounds for the symbols in (7.7) and (7.8). These can be obtained as in theprevious three sections. In particular, using the bounds (A.7) and (A.18), it is not hard toverify that

‖qk,k1,k2ε1,ε2 ‖S∞ . 2−k2/21X (k, k1, k2)1[−2 log2(2+t)−10,0](k). (7.11)

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46 ALEXANDRU D. IONESCU AND FABIO PUSATERI

This is a somewhat rougher bound than what actually holds true, but it will be sufficient forour estimates. Furthermore, using also∥∥∥ |ξ|1/2ηχ(η, ξ − η)

|ξ|3/2 − |ξ − η|3/2 − ε2|η|3/2ϕk(ξ)ϕk1(ξ − η)

∥∥∥S∞. 1,

one can verify that

‖|η|1/2qε1,ε2(ξ, η)ϕk(ξ)ϕk1(ξ − η)‖S∞ . 1[−2 log2(2+t)−10,0](k). (7.12)

We also have

‖|η|1/2qε1ε2(ξ, η)ϕk1(ξ − η)ϕk2(η)‖S∞ . 1 (7.13)

Similarly, using again (A.7) and (A.18) one can see that

‖rk,k1,k2ε1,ε2 ‖S∞ . 2−k1/21X (k, k1, k2)1[−2 log2(2+t)−10,0](k). (7.14)

and

‖|ξ − η|1/2rε1,ε2ϕk1(ξ − η)ϕk2(η)‖S∞ . 1. (7.15)

The proof of (7.9) can then be done in a similar fashion to what was done before in section5 in the proof of Lemma 5.2, by using the bounds (7.11)-(7.15) above.

7.3. Proof of Lemma 7.3. As in section 5.4, we use the definition of the quadratic energy(7.1), the equation (7.3)-(7.4), the formulas for the cubic energies (7.6)-(7.8), and a symmetriza-tion argument like the one performed for the term (4.17) and leading to (4.25), to calculate

d

dt

(E

(2)Sulow

+ E(3)Sulow

)=

1

2πL1 +

1

2π<L2 +

1

4π2<∑?

(L3,ε1ε2 + L4,ε1ε2 + L5,ε1ε2 + L6,ε1ε2

)+

1

4π2<∑?

(L7,ε1ε2 + L8,ε1ε2 + L9,ε1ε2 + L10,ε1ε2

)+R,

where

L1 =

∫R

∣∣Su(ξ)∣∣2 (1 + t)P ′((1 + t)2|ξ|)ϕ2

≤−10(ξ) dξ,

L2 =

∫RSu(ξ)F

(∂tSu− iΛSu

)(ξ)ϕ≤3((1 + t)2ξ)|ξ|−1P((1 + t)2|ξ|)ϕ2

≤−10(ξ) dξ,

L3,ε1ε2 =

∫R×R|ξ|−1/2Su(ξ) ∂tqε1ε2(ξ, η)uε1(ξ − η)Suε2(η) dξdη,

L4,ε1ε2 =

∫R×R|ξ|−1/2F

(∂tSu− iΛSu

)(ξ) qε1ε2(ξ, η)uε1(ξ − η)Suε2(η) dξdη,

L5,ε1ε2 =

∫R×R|ξ|−1/2Su(ξ) qε1ε2(ξ, η)F

(∂tuε1 − iε1Λuε1

)(ξ − η)Suε2(η) dξdη,

L6,ε1ε2 =

∫R×R|ξ|−1/2Su(ξ) qε1ε2(ξ, η)uε1(ξ − η)F

(∂tSuε2 − iε2ΛSuε2

)(η) dξdη,

(7.16)

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WATER WAVES WITH SURFACE TENSION 47

L7,ε1ε2 =

∫R×R|ξ|−1/2Su(ξ) ∂trε1ε2(ξ, η)uε1(ξ − η)uε2(η) dξdη,

L8,ε1ε2 =

∫R×R|ξ|−1/2F

(∂tSu− iΛSu

)(ξ) rε1ε2(ξ, η)uε1(ξ − η)uε2(η) dξdη,

L9,ε1ε2 =

∫R×R|ξ|−1/2Su(ξ) rε1ε2(ξ, η)F

(∂tuε1 − iε1Λuε1

)(ξ − η)uε2(η) dξdη,

L10,ε1ε2 =

∫R×R|ξ|−1/2Su(ξ) rε1ε2(ξ, η)uε1(ξ − η)F

(∂tuε2 − iε2Λuε2

)(η) dξdη.

(7.17)

The remainder R satisfies |R(t)| . ε41(1 + t)−1+8p0 , in view of (7.5).

Recall that under the assumption (3.45) we have

‖ϕ≥4((1 + t)2ξ)|ξ|−1/2u(t, ξ)‖L2 . ε1(1 + t)p0 ,

‖ϕ≤4((1 + t)2ξ)|ξ|−1/2+p1 u‖L2 . ε1(1 + t)p0−2p1 ,(7.18)

and

‖ϕ≥4((1 + t)2ξ)|ξ|−1/2Su(t, ξ)‖L2 . ε1(1 + t)4p0 ,

‖ϕ≤4((1 + t)2ξ)|ξ|−1/2+p1Su(t, ξ)‖L2 . ε1(1 + t)4p0−2p1 .(7.19)

7.3.1. Estimate of L1. Using (2.22) the definition of E(2)Sulow

in (7.1), we see that

1

2π|L1| ≤

1

∫R

∣∣Su(t, ξ)∣∣2 20p1

(1 + t)|ξ|P((1 + t)2|ξ|)ϕ2

≤−10(ξ) dξ ≤ 20p1E(2)Sulow

(t)(1 + t)−1,

as desired.

7.3.2. Estimate of L2. Notice that the integrand in L2 is supported on a region where |ξ| .(1 + t)−2, so that we can use the bound in (C.37), and (7.19), to obtain

|L2| .∑

k∈Z, 2k−10≤(1+t)−2

2k(−1+2p1)‖P ′kSu(t)‖L2‖P ′k(∂t − iΛ)Su(t)‖

L2(1 + t)4p1

.∑

k∈Z, 2k−10≤(1+t)−2

2k(−1+2p1)‖P ′kSu(t)‖L2ε212k/2(1 + t)−1+4p0+4p1

. ε1(1 + t)−1+8p0+2p1ε21

∑k∈Z, 2k−10≤(1+t)−2

2p1k

. ε31(1 + t)−1+8p0 .

7.3.3. Estimate of L3,ε1ε2. Looking at the definition of qε1ε2 in (7.7), and using (7.11), one cansee that

‖(∂tqε1ε2)k,k1,k2‖S∞ . (1 + t)−12−k2/21X (k, k1, k2)1[−2 log2(2+t)−10,−2 log2(2+t)+10](k).

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48 ALEXANDRU D. IONESCU AND FABIO PUSATERI

We can then use Lemma 2.1 to estimate, for every ε1, ε2 ∈ +,−,|L3,ε1ε2 | . L3,1 + L3,2,

L3,1 := 〈t〉−1∑

(k,k1,k2)∈X , k1≤k2+10, 2k≈〈t〉−2

2−k2/22−k/2‖P ′kSu‖L2‖P ′k1u‖L∞‖P′k2Su‖L2 ,

L3,2 := 〈t〉−1∑

(k,k1,k2)∈X , k1≥k2+10, 2k≈〈t〉−2

2−k2/22−k/2‖P ′kSu‖L2‖P ′k1u‖L2‖P ′k2Su‖L∞ .

Using (7.19) and the L∞ bound in (3.45) we see that

L3,1 . ε31〈t〉−1+4p0

∑k1,k2∈Z, 2k2+20≥〈t〉−2

2k1/102−N2 max(k1,0)〈t〉−1/22−k2/2‖P ′k2Su‖L2 . ε31〈t〉−1.

For the second term we can use (7.19) and the inequality

‖PlSu‖L∞ . 2(1−p1)lε1(1 + t)4p0−2p1 , if 2l . (1 + t)−2, (7.20)

to obtain

L3,2 . ε31〈t〉−1+10p0

∑k1,k2∈Z, 2k2.〈t〉−2

2−k2/2 min(2(1/2−p1)k1 , 2−N0 max(k1,0))2(1−p1)k2 . ε3

1〈t〉−4/3.

The desired bound |L3,ε1ε2 | . ε31(1 + t)−1+8p0 follows.

7.3.4. Estimate of L4,ε1ε2. To deal with the term L4,ε1ε2 we first use Lemma 2.1(ii) to estimate

|L4,ε1ε2 | . L4,1 + L4,2 + L4,3,

L4,1 :=∑

(k,k1,k2)∈X , 2k2+10≤(1+t)−2

‖qk,k1,k2ε1ε2 ‖S∞

2−k/2‖P ′k(∂t − iΛ)Su‖L2‖P ′k1u‖L2‖P ′k2Su‖L∞ ,

L4,2 := ‖ϕ≥−20((1 + t)2η)|η|−1/2Su(η)‖L2

∑k,k1∈Z, |k−k1|≤10

‖|η|1/2qε1ε2(ξ, η)ϕk(ξ)ϕk1(ξ − η)‖S∞

× 2−k/2‖P ′k(∂t − iΛ)Su‖L2‖P ′k1u‖L∞ ,

L4,3 :=∑

(k,k1,k2)∈X , k2+10≥k1

‖qk,k1,k2ε1ε2 ‖S∞

2−k/2‖P ′k(∂t − iΛ)Su‖L2‖P ′k1u‖L∞‖P′k2Su‖L2 .

Using the bounds (7.11), (C.37), (3.45), and (7.20), we see that

L4,1 .∑

(k,k1,k2)∈X , 2k2≤〈t〉−2, |k−k1|≤10

2−k2/2ε21〈t〉−1/2+4p0ε12(1/2−p1)k12−N0 max(k1,0)〈t〉p0

×ε12(1−p1)k2〈t〉4p0 . ε41〈t〉−1.

Using (7.19), the symbol bound (7.12), and (C.37), we obtain

L4,2 . ε1〈t〉4p0∑

k,k1∈Z, |k−k1|≤10

ε21〈t〉−1/2+4p0ε12k1/102−max(k1,0)〈t〉−1/2 . ε4

1〈t〉−1+8p0 .

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WATER WAVES WITH SURFACE TENSION 49

To estimate the last term we use (7.19), (7.11), and (C.37), and see that

L4,3 .∑

(k,k1,k2)∈X , 〈t〉−2≤2k≤210, k2+10≥k1

ε21〈t〉−1/2+4p0

(2k/2 + 〈t〉−1/2)

×ε12k1/102−max(k1,0)〈t〉−1/2ε1〈t〉4p0 . ε41〈t〉−1+8p0 .

The desired bound |L4,ε1ε2 | . ε31(1 + t)−1+8p0 follows.

7.3.5. Estimate of L5,ε1ε2. We first use Lemma 2.1(ii) to bound

|L5,ε1ε2 | . L5,1 + L5,2 + L5,3,

L5,1 :=∑

(k,k1,k2)∈X , 2k2+10≤(1+t)−2

‖qk,k1,k2ε1ε2 ‖S∞

2−k/2‖P ′kSu‖L2‖P ′k1(∂t − iΛ)u‖L2‖P ′k2Su‖L∞ ,

L5,2 := ‖ϕ≥−20((1 + t)2η)|η|−1/2Su(η)‖L2

∑k,k1∈Z, |k−k1|≤10

‖|η|1/2qε1ε2(ξ, η)ϕk(ξ)ϕk1(ξ − η)‖S∞

× 2−k/2‖P ′kSu‖L2‖P ′k1(∂t − iΛ)u‖L∞,

L5,3 := ‖ϕ≥0((1 + t)2ξ)|ξ|−1/2Su(ξ)‖L2

∑k1,k2∈Z, k2+10≥k1

‖|η|1/2qε1ε2(ξ, η)ϕk1(ξ − η)ϕk2(η)‖S∞

× ‖P ′k1(∂t − iΛ)u‖L∞

2−k2/2‖P ′k2Su‖L2 .

Using the symbol bound (7.11), (C.37), (3.45), and (7.20), we get:

L5,1 .∑

k,k1,k2∈Z, 2k2+10≤〈t〉−2, |k−k1|≤10

2−k2/2ε1〈t〉4p0ε212k1/22−max(k1,0)〈t〉−1/2+p0

×ε12(1−p1)k2〈t〉4p0−2p1 . ε41〈t〉−1.

Using (7.19), the symbol bound (7.12), and (C.35), we obtain

L5,2 . ε1〈t〉4p0∑

k,k1∈Z, |k−k1|≤10

ε1〈t〉4p0ε212k1/22−max(k1,0)〈t〉−1 . ε4

1〈t〉−1+8p0 .

Similarly, using (7.19), (7.13), and (C.35), we see that

L5,3 . ε1〈t〉4p0∑

k1,k2∈Z, k2+10≥k1

ε212k1/22−max(k1,0)〈t〉−1ε12−p1k2〈t〉4p0 . ε4

1〈t〉−1+8p0 .

The desired bound |L5,ε1ε2 | . ε31(1 + t)−1+8p0 follows.

7.3.6. Estimate of L6,ε1ε2. Using Lemma 2.1(ii) we can bound

|L6,ε1ε2 | . L6,1 + L6,2,

L6,1 :=∑

k,k1,k2∈Z, 2k2≤〈t〉−2

‖qk,k1,k2ε1ε2 ‖S∞

2−k/2‖P ′kSu‖L2‖P ′k1u‖L2‖P ′k2(∂t − iΛ)Su‖L∞,

L6,2 := ‖ϕ≥0((1 + t)2ξ)|ξ|−1/2Su(ξ)‖L2

∑k1,k2∈Z, 2k2≥〈t〉−2

‖|η|1/2qε1ε2(ξ, η)ϕk1(ξ − η)ϕk2(η)‖S∞

× ‖P ′k1u‖L∞2−k2/2‖P ′k2(∂t − iΛ)Su‖L2 .

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50 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Using (7.11), (7.18)–(7.19), and (C.37), we have

L6,1 .∑

k,k1,k2∈Z, 2k2≤〈t〉−2, k2−10≤k≤10

2−k2/2ε1〈t〉4p0ε12(1/2−p1)k12−max(k1,0)〈t〉p0

×2k2/2ε212k2/2〈t〉−1/2+4p0 . ε4

1〈t〉−1.

Using (7.19), (7.13), (3.45), and (C.37), we have

L6,2 . ε1〈t〉4p0∑

k1,k2∈Z, 2k2≥〈t〉−2

ε212k1/102−max(k1,0)〈t〉−1/2

× ε21〈t〉−1/2+4p0(2k2/2 + 〈t〉−1/2)2−max(k2,0) . ε4

1〈t〉−1+8p0 .

The desired bound |L6,ε1ε2 | . ε31(1 + t)−1+8p0 follows.

7.3.7. Estimate of Lj,ε1ε2, j = 7, . . . , 10. Observe that the terms Lj,ε1ε2 , for j = 7, . . . , 10 in(7.17) are easier to estimate than the terms Lj,ε1ε2 , for j = 3, . . . , 6 in (7.16). This is becausethe bounds for the symbols are essentially the same, see (7.11)-(7.15), but we have strongerinformation on u than on Su. Therefore, the integrals Lj,ε1ε2 , for j = 7, . . . , 10 can be treatedsimilarly to the ones that we have just estimated. We can then conclude the desired bound ofε3

1(1 + t)−1+8p0 for the evolution of ESulow . This gives Lemma 7.3, and hence Proposition 7.1.

8. Decay estimates

To prove decay we return to the Eulerian variables (h, φ) and prove the following:

Proposition 8.1. With U = |∂x|h− i|∂x|1/2φ as in Proposition 2.4, we have

supt∈[0,T ]

(1 + t)1/2∑k∈Z

(2−k/10 + 2N2k

)‖PkU(t)‖L∞ . ε0. (8.1)

Assuming this proposition, we can complete the proof of the main Proposition 2.4.

Proof of Proposition 2.4. It follows from Propositions 4.1, 5.1, 6.1, and 7.1 that

〈t〉−p0K′I(t) + 〈t〉−4p0K′S(t)+ . ε0

for any t ∈ [0, T ]. It follows from (3.17) that u − U ∈ O2,0 is a quadratic expression thatdoes not lose derivatives and vanishes at frequencies ≤ 2−10. The desired conclusion (2.28)follows.

The proof of Proposition 8.1 will be given through a series of steps below. The strategyfollows the general approach of [32, 34].

8.1. Set up. Recall that for any suitable multiplier m : Rd → C we define the associatedbilinear and trilinear operators M by the formulas

F[M(f, g)

](ξ) =

1

∫Rm(ξ, η)f(ξ − η)g(η) dη,

F[M(f, g, h)

](ξ) =

1

4π2

∫R×R

m(ξ, η, σ)f(ξ − η)g(η − σ)h(σ) dηdσ.

In view of Lemma C.5, the variable U satisfies the equation

(∂t − iΛ)U = QU + CU +R≥4, Λ = |∂x|3/2, (8.2)

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WATER WAVES WITH SURFACE TENSION 51

where:

• The quadratic nonlinear terms are

QU =∑

(ε1ε2)∈(++),(+−),(−−)

Qε1ε2(Uε1 , Uε2), (8.3)

where U+ = U , U− = U , and the operators Q++, Q+−, Q−− are defined by the symbols

q++(ξ, η) :=i|ξ|(ξη − |ξ||η|)8|η|1/2|ξ − η|

+i|ξ|(ξ(ξ − η)− |ξ||ξ − η|)

8|η||ξ − η|1/2+i|ξ|1/2(η(ξ − η) + |η||ξ − η|)

8|η|1/2|ξ − η|1/2,

q+−(ξ, η) := − i|ξ|(ξη − |ξ||η|)4|η|1/2|ξ − η|

+i|ξ|(ξ(ξ − η)− |ξ||ξ − η|)

4|η||ξ − η|1/2− i|ξ|1/2(η(ξ − η) + |η||ξ − η|)

4|η|1/2|ξ − η|1/2,

q−−(ξ, η) := − i|ξ|(ξη − |ξ||η|)8|η|1/2|ξ − η|

− i|ξ|(ξ(ξ − η)− |ξ||ξ − η|)8|η||ξ − η|1/2

+i|ξ|1/2(η(ξ − η) + |η||ξ − η|)

8|η|1/2|ξ − η|1/2.

(8.4)

• The cubic terms have the form

CU := M++−(U,U, U) +M+++(U,U, U) +M−−+(U,U, U) +M−−−(U,U,U), (8.5)

with purely imaginary symbols mι1ι2ι3 such that∥∥F−1[mι1ι2ι3(ξ, η, σ) · ϕk(ξ)ϕk1(ξ − η)ϕk2(η − σ)ϕk3(σ)

]∥∥L1 . 2k/22max(k1,k2,k3) (8.6)

for all (ι1ι2ι3) ∈ (+ +−), (−−+), (+ + +), (−−−). Moreover, with d1 = 1/8,

m++−(ξ, 0,−ξ) = id1|ξ|3/2. (8.7)

• R≥4 is a quartic remainder satisfying

‖R≥4‖L2 + ‖SR≥4‖L2 . ε41〈t〉−5/4. (8.8)

Moreover

CU +R≥4 ∈ |∂x|1/2O3,−1. (8.9)

8.2. The “semilinear” normal form transformation. We follow the classical normal formapproach of Shatah [43] to define a modified variable which is a quadratic perturbation of uand satisfies a cubic equation. Let

v := U +M++(U,U) +M+−(U,U) +M−−(U,U), (8.10)

where, for any (ε1, ε2) ∈ (++), (+−), (−−), the bilinear operators Mε1ε2 are defined by themultipliers

mε1ε2(ξ, η) := −i qε1ε2(ξ, η)

|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2. (8.11)

A direct computation shows that v solves the equation

(∂t − iΛ)v =∑∗

′[Mε1ε2((LU)ε1 , Uε2) +Mε1ε2(Uε1 , (LU)ε2)

]+ CU +R≥4 (8.12)

where∑∗′ :=

∑(ε1,ε2)∈(++),(+−),(−−) and L := (∂t − iΛ).

We now prove several bounds on the new variable v.

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52 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Lemma 8.2. Let v be defined by (8.10)-(8.11). Then for any t ∈ [0, T ] and k ∈ Z we have

‖Pk(U(t)− v(t))‖L∞ . ε21 min

(2k/2, 2−(N2−1/2)k

)〈t〉−3/4+2p0 , (8.13)

‖Pk(U(t)− v(t))‖L2 . ε21 min(2k/2, 2−(N0−1/2)k)〈t〉−1/4+3p0 , (8.14)

‖PkS(U(t)− v(t))‖L2 . ε21 min

(2k/2, 2−(N1−1/2)k

)〈t〉−1/4+6p0 . (8.15)

Furthermore, we have

‖Pk(U(t)− v(t))‖L2 . ε21〈t〉−1/2+3p0 min

(2k/2, 2−(N2−1/2)k

). (8.16)

The bounds (8.13)-(8.15) show that v and u have the same relevant norms, and that all thea priori assumptions on u transfer without significant losses to v. The bound (8.16) is a variantof the L2 bound (8.14) which provides more decay in time, but less decay at high frequencies.This will be used later on to bound quartic remainder terms.

Remark 8.3. Observe that (8.13), (8.14) and Sobolev embedding, imply

‖Pk(U(t)− v(t))‖L∞ . ε21〈t〉−1/2 min

(2−(N2+1)k, 2k/8

). (8.17)

This shows that in order to obtain (8.1) is suffices to show

supt∈[0,T ]

〈t〉1/2∑k∈Z

(2−k/10 + 2N2k

)‖Pkv(t)‖L∞ . ε0. (8.18)

Proof of Lemma 8.2. In view of (8.10), we see that we have to estimate the bilinear termsMε1ε2 . Notice that the formulas (8.4) and (8.11) show easily that

‖qk,k1,k2ε1ε2 ‖S∞. 2k2min(k,k1,k2)/21X (k, k1, k2),

‖mk,k1,k2ε1ε2 ‖

S∞. 2k/22−min(k1,k2)/21X (k, k1, k2).

(8.19)

The a priori bounds (2.25) show that, for any l ∈ Z and t ∈ [0, T ],

‖PlU(t)‖L2 . ε1〈t〉p0 min(2l(1/2−p1), 2−N0l

),

‖PlU(t)‖L∞ . ε1〈t〉−1/2 min(2l/10, 2−N2l),

‖PlSU(t)‖L2 . ε1〈t〉4p0 min(2(1/2−p1)l, 2−N1l

).

(8.20)

Therefore, using Lemma 2.1(ii), (8.19), and (8.20), we estimate, for any k ∈ Z and t ∈ [0, T ],

‖PkMε1ε2(Uε1 , Uε2)‖L∞ .∑k1≤k2

‖mk,k1,k2ε1ε2 ‖

S∞‖P ′k1U‖L∞‖P

′k2U‖L∞

. ε21〈t〉−1

∑k1≤k2, 2k1≥〈t〉−1/2

1X (k, k1, k2)2(k−k1)/2 min(2−N2k1 , 2k1/10) min(2−N2k2 , 2k2/10)

+ ε21〈t〉−1/2+p0

∑k1≤k2, 2k1≤〈t〉−1/2

1X (k, k1, k2)2(k−k1)/22k1/22(1/2−p1)k1 min(2−N2k2 , 2k2/10)

. ε21 min

(2k/2, 2−(N2−1/2)k

)〈t〉−3/4+2p0 ,

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WATER WAVES WITH SURFACE TENSION 53

which suffices. We proceed similarly to obtain (8.14):

‖PkMε1ε2(Uε1 , Uε2)‖L2 .∑k1≤k2

‖mk,k1,k2ε1ε2 ‖

S∞‖P ′k1U‖L∞‖P

′k2U‖L2

. ε21〈t〉−1/2+p0

∑k1≤k2, 2k1≥〈t〉−1/2

1X (k, k1, k2)2(k−k1)/2 min(2−N0k2 , 2(1/2−p1)k2)

+ ε21〈t〉2p0

∑k1≤k2, 2k1≤〈t〉−1/2

1X (k, k1, k2)2(k−k1)/22(1−p1)k1 min(2−N0k2 , 2(1/2−p1)k2)

. ε21 min

(2k/2, 2−(N0−1/2)k

)〈t〉−1/4+3p0 .

Using (2.5)-(2.6) we have

SMε1ε2(f, g) = Mε1ε2(Sf, g) +Mε1ε2(f, Sg) + Mε1ε2(f, g)

where Mε1ε2 is the operator associated to mε1,ε2(ξ, η) = −(ξ∂ξ + η∂η)mε1ε2(ξ, η). It is not hardto verify that the symbols mε1,ε2 satisfy the same bounds (8.19) as the symbols mε1,ε2 and,therefore, estimating as above

‖Mε1ε2(Uε1 , Uε2)‖L2 . ε21 min

(2k/2, 2−(N0−1/2)k

)〈t〉−1/4+3p0 .

Thus, to prove (8.15) it suffices to estimate Mε1ε2(SUε1 , Uε2) and Mε1ε2(Uε1 , SUε2). As in theproof of (8.14), we use Lemma 2.1(ii), followed by (8.19), and (8.20), to obtain

‖PkMε1ε2(SUε1 , Uε2)‖L2

.∑

k1,k2∈Z‖mk,k1,k2

ε1ε2 ‖S∞

min(‖P ′k1SU‖L2‖P ′k2U‖L∞ , ‖P

′k1SU‖L∞‖P

′k2U‖L2

). (8.21)

We use the bound ‖P ′k1SU‖L2‖P ′k2U‖L∞ when k2 ≤ k1 or when k1 ≤ k2 and 2k1 ≥ 〈t〉1/2.

We use the bound ‖P ′k1SU‖L∞‖P′k2U‖

L2 when k1 ≤ k2 and 2k1 ≤ 〈t〉1/2. It follows that the

left-hand side of (8.21) is bounded by

Cε21〈t〉−1/2+4p0

∑k2≤k1, 2k2≥〈t〉−1/2

1X (k, k1, k2)2(k−k2)/22−N2k+2 min(2−N1k1 , 2(1/2−p1)k1)

+ Cε21〈t〉5p0

∑k2≤k1, 2k2≤〈t〉−1/2

1X (k, k1, k2)2(k−k2)/22(1−p1)k2 min(2−N1k1 , 2(1/2−p1)k1)

+ Cε21〈t〉−1/2+4p0

∑k1≤k2, 2k1≥〈t〉−1/2

1X (k, k1, k2)2(k−k1)/22−N2k+2 min(2−N1k1 , 2(1/2−p1)k1)

+ Cε21〈t〉5p0

∑k1≤k2, 2k1≤〈t〉−1/2

1X (k, k1, k2)2(k−k1)/22−N2k+2 2(1−p1)k1

. ε21 min

(2k/2, 2−(N1−1/2)k

)〈t〉−1/4+6p0 .

This completes the proof for ‖PkMε1ε2(SUε1 , Uε2)‖L2 . The proof for ‖PkMε1ε2(Uε1 , SUε2)‖L2 issimilar, due to the symmetric bounds on the symbols mε1ε2 .

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54 ALEXANDRU D. IONESCU AND FABIO PUSATERI

To prove (8.16), which gives better time decay than (8.14), we use again Lemma 2.1(ii), thesymbol bounds (8.19), and the a priori bounds on u in (8.20),

‖PkMε1ε2(Uε1 , Uε2)‖L2 .∑k1≤k2

‖mk,k1,k2ε1ε2 ‖

S∞min

(‖P ′k1U‖L2‖P ′k2U‖L∞ , ‖P

′k1U‖L∞‖P

′k2U‖L2

). ε2

1〈t〉−1/2+p0∑

k1≤k2, 2k1≥〈t〉−4

1X (k, k1, k2)2(k−k1)/2 min(2(1/2−p1)k1 , 2−N0k1) min(2−N2k2 , 2k2/10)

+ ε21〈t〉2p0

∑k1≤k2, 2k1≤〈t〉−4

1X (k, k1, k2)2(k−k1)/22(1−p1)k1 min(2−N0k2 , 2(1/2−p1)k2)

. ε21 min

(2k/2, 2−(N2−1/2)k

)〈t〉−1/2+3p0 .

This completes the proof of the lemma.

8.3. The profile f . For t ∈ [0, T ] we define the profile of the solution of (8.12) as

f(t) := e−itΛv(t). (8.22)

In the next proposition we summarize the main properties of the function f .

Proposition 8.4 (Bounds for the profile). With v and f are defined as above, we have

eitΛ∂tf = (∂t − iΛ)v = N ′

N ′ :=∑?

′Mε1ε2((LU)ε1 , Uε2) +Mε1ε2(Uε1 , (LU)ε2) + CU +R≥4,

(8.23)

where the bilinear operators Mε1ε2 are defined via (8.11), and CU and R≥4 satisfy (8.5)–(8.8).Moreover, for any t ∈ [0, T ] and k ∈ Z, we have the estimates∥∥Pk(eitΛf(t))

∥∥L∞. ε1 min(2k/10, 2−N2k)〈t〉−1/2, (8.24)

‖Pkf(t)‖L2 . ε0〈t〉6p0 min(2(1/2−p1)k, 2−(N0−1/2)k

), (8.25)

‖Pk(x∂xf(t))‖L2 . ε0〈t〉6p0 min(2(1/2−p1)k, 2−(N1−1/2)k

). (8.26)

Proof. The equation (8.23) follows from the definition (8.22) and the equation (8.12). The L∞

bound (8.24) follows from (8.17) and (8.20). The L2 bound (8.25) follows from the energyestimates in Propositions 4.1 and 5.1, and the bounds (8.14).

To prove (8.26) we start from the identity

Sv = eitΛ(x∂xf) + (3/2)teitΛ(∂tf),

which is a consequence of the commutation identity [S, eitΛ] = 0. Therefore, for any t ∈ [0, T ]and k ∈ Z,

‖Pk(x∂xf(t))‖L2 . ‖Pk(Sv(t))‖L2 + (1 + t)‖Pk(∂t − iΛ)v(t)‖L2 . (8.27)

Using Proposition 6.1 and Proposition 7.1, we know, in particular, that

‖PkSU(t)‖L2 . ε0 min(2(1/2−p1)k, 2−N1k)〈t〉4p0 , (8.28)

for all k ∈ Z. Together with (8.15) and (8.27), this gives

‖Pk(x∂xf(t))‖L2 . 〈t〉‖Pk(∂t − iΛ)v(t)‖L2 + ε21

(2k/2, 2−(N1−1/2)k

)〈t〉−1/4+6p0

+ε0 min(2(1/2−p1)k, 2−N1k)〈t〉4p0 .

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WATER WAVES WITH SURFACE TENSION 55

It then suffices to show

‖Pk(∂t − iΛ)v(t)‖L2 . 〈t〉−1+6p0ε21

(2(1/2−p1)k, 2−(N1−1/2)k

). (8.29)

Since CU +R≥4 ∈ |∂x|1/2O3,−1, see (8.9), its contribution is already bounded by the right-hand side above, according to the definition of O3,α, see (C.3). To complete the proof of (8.29)it suffices to estimate Mε1ε2(Uε1 , (LU)ε2) and Mε1ε2((LU)ε1 , Uε2).

It follows from (8.9) and the symbol bounds on qε1ε2 in (8.19) that

‖PlLU(t)‖L2 . ε21 min(2l/2, 2−(N0−2)l)〈t〉−1/2+p0 ,

‖PlLU(t)‖L∞ . ε21 min(2l/2, 2−(N2−2)l)〈t〉−1.

(8.30)

Then, using Lemma 2.1(ii) with the symbol bounds (8.19), (8.20) and (8.30), we can estimate

‖PkMε1ε2(Uε1(t), (LU)ε2(t))‖L2

.∑

k1,k2∈Z‖mk,k1,k2

ε1ε2 ‖S∞

min(‖P ′k1U(t)‖

L2‖P ′k2LU(t)‖L∞, ‖P ′k1U(t)‖

L∞‖P ′k2LU(t)‖

L2

). (8.31)

We use the bound ‖P ′k1U(t)‖L2‖P ′k2LU(t)‖

L∞when (k1 ≤ k2 and 2k1 ≥ 〈t〉−4) or when (k2 ≤ k1

and 2k2 ≤ 〈t〉−4). We use the bound ‖P ′k1U(t)‖L∞‖P ′k2LU(t)‖

L2 when (k1 ≤ k2 and 2k1 ≤ 〈t〉−4)

or when (k2 ≤ k1 and 2k2 ≥ 〈t〉−4). It follows that the left-hand side of (8.31) s bounded by

Cε21〈t〉−1+p0

∑k1≤k2, 2k1≥〈t〉−4

1X (k, k1, k2)2(k−k1)/2 min(2(1/2−p1)k1 , 2−N0k1) min(2k2/2, 2−(N2−2)k2)

+ Cε21〈t〉−1/2+2p0

∑k1≤k2, 2k1≤〈t〉−4

1X (k, k1, k2)2(k−k1)/22(1−p1)k1 min(2k2/2, 2−(N0−2)k2)

+ Cε21〈t〉−1+p0

∑k2≤k1, 2k2≥〈t〉−4

1X (k, k1, k2)2(k−k2)/22−N2k+1 min(2k2/2, 2−k2)

+ Cε21〈t〉−1/2+2p0

∑k2≤k1, 2k2≤〈t〉−4

1X (k, k1, k2)2(k−k2)/22−N2k+1 2k2

. 〈t〉−1+6p0ε21

(2(1/2−p1)k, 2−(N1−1/2)k

).

The bound on ‖PkMε1ε2((LU)ε1(t), Uε2(t))‖L2 is similar. This completes the proof of (8.29)and the proposition.

8.4. The Z-norm and proof of Proposition 8.1. For any function h ∈ L2(R) let

‖h‖Z :=∥∥(|ξ|1/10 + |ξ|N2+1/2

)h(ξ)

∥∥L∞ξ. (8.32)

Proposition 8.5. Let f be defined as in (8.22) and assume that for T ′ ∈ [0, T ]

supt∈[0,T ′]

‖f(t)‖Z ≤ ε1. (8.33)

Then

supt∈[0,T ′]

‖f(t)‖Z . ε0. (8.34)

We now show how to prove Proposition 8.1 using Proposition 8.5 above.

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56 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Proof of Proposition 8.1. Define z(t) := ‖f(t)‖Z and notice that z : [0, T ]→ R+ is a continuousfunction. We show first that

z(0) . ε0. (8.35)

Indeed, using the definitions and Lemma 2.3 we get

z(0) . supk∈Z

(2k/10 + 2(N2+1/2)k)‖Pkv(0)‖L∞

. supk∈Z

(2k/10 + 2(N2+1/2)k)2−k/2‖P ′kv(0)‖1/2

L2

(2k‖∂P ′kv(0)‖

L2 + ‖P ′kv(0)‖L2

)1/2.

Thanks to (8.14)-(8.16) with t = 0, and the initial data assumptions (3.47), we have

‖P ′kv(0)‖L2 . ε0 min(2(1/2−p1)k, 2−(N0−1/2)k),

2k‖∂P ′kv(0)‖L2 . ε0 min(2(1/2−p1)k, 2−(N1−1/2)k),

so that (8.35) follows, using also (N0 +N1)/2 ≥ N2 + 1, see (2.23).We apply now Proposition 8.5. By continuity, z(t) . ε0 for any t ∈ [0, T ], provided that ε0 is

sufficiently small and ε0 ε1 ≤ ε2/30 1 as in (2.23). Therefore, for any k ∈ Z and t ∈ [0, T ],

(2k/10 + 2(N2+1/2)k)‖Pkf(t)‖L∞ . ε0. (8.36)

Recall that we aim to prove, for all t ∈ [1, T ], the decay bound

supt∈[0,T ]

〈t〉1/2∑k∈Z

(2−k/10 + 2N2k

)‖Pkv(t)‖L∞ . ε0, (8.37)

which, as already observed, implies (8.1) via (8.17) (the bound for t ∈ [0, 1] is a consequenceof Propositions 4.1 and 5.1). Also, observe that Lemma 2.2 applied to v = eitΛf gives

‖Pkv(t)‖L∞ . t−1/22k/4‖P ′kf(t)‖

L∞ + t−3/52−2k/5[2k‖∂P ′kf(t)‖

L2 + ‖P ′kf(t)‖L2

](8.38)

and‖Pkv(t)‖L∞ . t

−1/22k/4‖P ′kf(t)‖L1 . (8.39)

Recall that from (8.25) and (8.26) we have

2k‖∂P ′kf(t)‖L2 + ‖P ′kf(t)‖

L2 . ε0〈t〉6p02(1/2−p1)k (8.40)

We can then use (8.36) and (8.40) in (8.38), to obtain

‖Pkv(t)‖L∞ . t−1/22k/4

ε0

2k/10 + 2(N2+1/2)k+ t−3/52−2k/5ε0〈t〉6p02(1/2−p1)k,

which is enough to show that∑k∈Z, 〈t〉−100p0≤2k≤〈t〉100p0

(2−k/10 + 2N2k

)‖Pkv(t)‖L∞ . ε0〈t〉−1/2. (8.41)

Combining (8.39), with Lemma 2.3, and (8.25)-(8.26) we see that

‖Pkv(t)‖L∞ . t−1/22k/4ε0(1 + t)6p0 min(2−kp1 , 2−(N0+N1)k/2). (8.42)

Using also (2.23) it follows that∑k∈Z, 2k≤〈t〉−100p0

2−k/10‖Pkv(t)‖L∞ +∑

k∈Z, 2k≥〈t〉100p02N2k‖Pkv(t)‖L∞ . ε0(1 + t)−1/2. (8.43)

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WATER WAVES WITH SURFACE TENSION 57

The estimates (8.41) and (8.43) give us (8.37), and conclude the proof of Proposition 8.1.

8.5. The equation for v and proof of Proposition 8.5. We now derive an equation for vwith cubic terms that only involve v itself. Recall that u solves (8.2)-(8.8), v solves (8.23), andthey are related via (8.10)-(8.11).

According to (8.3) and (8.4) we define

Qv :=∑∗

′Qε1ε2(vε1 , vε2) =

∑(ε1ε2)∈(++),(+−),(−−)

Qε1ε2(vε1 , vε2), (8.44)

where the operators Qε1,ε2 are defined by the symbols qε1ε2 . We use the notation∑??

:=∑

(ε1ε2ε3)∈(++−),(+++),(−−+),(−−−)

and rewrite (8.23) in the form

∂tf = e−itΛ(N ′′ +R′′≥4) (8.45)

where

N ′′ :=∑?

′Mε1ε2((Qv)ε1 , vε2) +Mε1ε2(vε1 , (Qv)ε2) +

∑??

Mε1ε2ε3(vε1 , vε2 , vε3) (8.46)

and

R′′≥4 :=∑?

′[Mε1ε2

((QU )ε1 , Uε2

)−Mε1ε2

((Qv)ε1 , vε2

)]+∑?

′[Mε1ε2

(Uε1 , (QU )ε2

)−Mε1ε2

(vε1 , (Qv)ε2

)]+∑??

[Mε1ε2ε3(Uε1 , Uε2 , Uε3)−Mε1ε2ε3(vε1 , vε2 , vε3)

]+R≥4

+∑?

′Mε1ε2

((CU +R≥4)ε1 , Uε2

)+Mε1ε2

(Uε1 , (CU +R≥4)ε2

).

(8.47)

The point of the above decomposition is to identify N ′′ as the main “cubic” part of the non-linearity, which can be expressed only in terms of v(t) = eitΛf(t). R′′≥4 can be thought of as aquartic remainder, due to the quadratic nature of u− v, see Lemma 8.2.

To analyze the equation (8.45), and identify the asymptotic logarithmic phase correction,

we need to distinguish among different types of interactions in the nonlinearity N ′′ . We write

N ′′ = C++− + C+++ + C−−+ + C−−−, (8.48)

where, recalling that the operators Q++,M++, Q−−,M−− are symmetric,

C++− = C++−(v, v, v) := 2M++(v,Q+−(v, v)) +M+−(Q++(v, v), v)

+M+−(v,Q+−(v, v)) + 2M−−(Q−−(v, v), v) +M++−(v, v, v),(8.49)

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58 ALEXANDRU D. IONESCU AND FABIO PUSATERI

C+++ = C+++(v, v, v) := 2M++(v,Q++(v, v)) +M+−(v,Q−−(v, v)) +M+++(v, v, v),

C−−+ = C−−+(v, v, v) := 2M++(v,Q−−(v, v)) +M+−(Q+−(v, v), v) +M+−(v,Q++(v, v))

+ 2M−−(Q+−(v, v), v) +M−−+(v, v, v),

C−−− = C−−−(v, v, v) := M+−(Q−−(v, v), v) + 2M−−(v,Q++(v, v)) +M−−−(v, v, v).

Notice that

Qε1ε2(g1, g2) = −Qε1ε2(g1, g2). (8.50)

Letting v+ = v, v− = v, we expand

Cι1ι2ι3(ξ) =i

4π2

∫R×R

cι1ι2ι3(ξ, η, σ)vι1(ξ − η)vι2(η − σ)vι3(σ) dηdσ (8.51)

for (ι1, ι2, ι3) ∈ (+ +−), (+ + +), (−−+), (−−−), where

ic++−(ξ, η, σ) : = 2m++(ξ, η)q+−(η, σ) +m+−(ξ, σ)q++(ξ − σ, ξ − η)

−m+−(ξ, η)q+−(η, η − σ)− 2m−−(ξ, σ)q−−(ξ − σ, ξ − η) +m++−(ξ, η, σ),

ic+++(ξ, η, σ) : = 2m++(ξ, η)q++(η, σ)−m+−(ξ, η)q−−(η, σ) +m+++(ξ, η, σ),

ic−−+(ξ, η, σ) : = 2m++(ξ, σ)q−−(ξ − σ, ξ − η) +m+−(ξ, ξ − η)q+−(η, η − σ)

−m+−(ξ, ξ − σ)q++(ξ − σ, ξ − η)− 2m−−(ξ, η)q+−(η, σ) +m−−+(ξ, η, σ),

ic−−−(ξ, η, σ) : = m+−(ξ, ξ − η)q−−(η, σ)− 2m−−(ξ, η)q++(η, σ) +m−−−(ξ, η, σ).(8.52)

Using the definitions of the quadratic symbols (8.11) and (8.4), we see that the cubic symbolscι1ι2ι3 are real-valued. Recalling the formulas

v+(ξ, t) = f(ξ, t)eit|ξ|3/2, v−(ξ, t) = f(ξ, t)e−it|ξ|

3/2,

we can rewrite

F(e−itΛN ′′(t)

)(ξ, t) =

i

4π2

[I++−(ξ, t) + I+++(ξ, t) + I−−+(ξ, t) + I−−−(ξ, t)

], (8.53)

where

Iι1ι2ι3(ξ) :=

∫R×R

eit(−|ξ|3/2+ι1|ξ−η|3/2+ι2|η−σ|3/2+ι3|σ|3/2)

× cι1ι2ι3(ξ, η, σ)f ι1(ξ − η)f ι2(η − σ)f ι3(σ) dηdσ

(8.54)

for (ι1, ι2, ι3) ∈ (+ +−), (+ + +), (−−+), (−−−). The formulas (8.45)-(8.47) become

(∂tf)(ξ, t) =i

4π2

[I++−(ξ, t)+I+++(ξ, t)+I−−+(ξ, t)+I−−−(ξ, t)

]+e−it|ξ|

3/2R′′≥4(ξ, t). (8.55)

In analyzing the formula (8.55), the main contribution comes from the stationary points ofthe phase functions (t, η, σ)→ tΨι1ι2ι3(ξ, η, σ), where

Ψι1ι2ι3(ξ, η, σ) := −|ξ|3/2 + ι1|ξ − η|3/2 + ι2|η − σ|3/2 + ι3|σ|3/2. (8.56)

More precisely, one needs to understand the contribution of the spacetime resonances, i.e., thepoints where

Ψι1ι2ι3(ξ, η, σ) = (∂ηΨι1ι2ι3)(ξ, η, σ) = (∂σΨι1ι2ι3)(ξ, η, σ) = 0.

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WATER WAVES WITH SURFACE TENSION 59

In our case, it can be easily verified that the only spacetime resonances (except for (ξ, η, σ) =(0, 0, 0)) correspond to (ι1ι2ι3) = (++−) and (ξ, η, σ) = (ξ, 0,−ξ). Moreover, the contributionfrom these points is not absolutely integrable in time, and we have to identify and eliminateits leading order term using a suitable logarithmic phase correction. More precisely, see also(C.59), we define, with d2 = −1/16, see (C.59),

c(ξ) := −8π|ξ|1/2

3c++−(ξ, 0,−ξ) = −8πd2|ξ|2

3=π|ξ|2

6,

L(ξ, t) :=c(ξ)

4π2

∫ t

0|f(ξ, s)|

2 1

s+ 1ds,

g(ξ, t) := eiL(ξ,t)f(ξ, t).

(8.57)

The formula (8.55) then becomes

(∂tg)(ξ, t) =i

4π2eiL(ξ,t)

[I++−(ξ, t) + c(ξ)

|f(ξ, t)|2

t+ 1f(ξ, t)

]+

i

4π2eiL(ξ,t)

[I+++(ξ, t) + I−−+(ξ, t) + I−−−(ξ, t)

]+ e−it|ξ|

3/2eiL(ξ,t)R′′≥4(ξ, t).

(8.58)

Notice that the phase L is real-valued. Therefore, to complete the proof of Proposition 8.5, itsuffices to prove the following main lemma:

Lemma 8.6. Recall the bounds (8.25)-(8.26) and the apriori assumption (8.33). Then, forany m ∈ 1, 2, . . . and any t1 ≤ t2 ∈ [2m − 2, 2m+1] ∩ [0, T ′], we have∥∥(|ξ|1/10 + |ξ|N2+1/2)[g(ξ, t2)− g(ξ, t1)]

∥∥L∞ξ. ε02−p0m. (8.59)

9. Proof of Lemma 8.6

In this section we provide the proof of Lemma 8.6. We first notice that the desired conclusioncan be easily proved for large and small enough frequencies. Indeed, for any t ∈ [2m−2, 2m+1]∩[0, T ′], and any |ξ| ≈ 2k with k ∈ Z and

k ∈ (−∞,−80p0m] ∪ [20p0m,∞),

we can use the interpolation inequality (2.20) and the bounds (8.25)-(8.26) to obtain(|ξ|1/10 + |ξ|N2+1/2

)|g(ξ, t)| . (2k/10 + 2(N2+1/2)k)

[2−k‖Pkf‖L2

(2k‖∂Pkf‖L2 + ‖Pkf‖L2

)]1/2. ε0〈t〉6p0 min

(2(1/10−p0)k, 2−k/2

),

. ε0〈t〉−p0 ,

having also used (N0 +N1)/2 ≥ N2 + 1, see (2.23).It remains to prove (8.59) in the intermediate range |ξ| ∈ [(1+ t)−80p0 , (1+ t)20p0 ]. For k ∈ Z

let f+k := Pkf and f−k := Pkf and, for any k1, k2, k3 ∈ Z, let

Iι1ι2ι3k1,k2,k3(ξ, t) :=

∫R×R

eit(−|ξ|3/2+ι1|ξ−η|3/2+ι2|η−σ|3/2+ι3|σ|3/2)

× cι1ι2ι3(ξ, η, σ)f ι1k1(ξ − η)f ι2k2(η − σ)f ι3k3(σ) dηdσ.

(9.1)

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60 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Using (8.25)-(8.26), (8.33) and Lemma 2.2 we know that for any t ∈ [0, T ′] and l ≤ 0

‖f±l (t)‖L2 + 2l‖∂f±l (t)‖

L2 . ε12(1/2−p0)l〈t〉6p0 ,

‖e±itΛf±l (t)‖L∞ . ε12l/10〈t〉−1/2,

‖f±l (t)‖L∞ . ε12−l/10,

(9.2)

whereas, for l ≥ 0, using also Lemma 2.3 and (2.19),

‖f±l (t)‖L2 . ε12−(N0−1/2)l〈t〉6p0 ,

‖f±l (t)‖L2 + 2l‖∂f±l (t)‖

L2 . ε12−(N1−1/2)l〈t〉6p0 ,

‖e±itΛf±l (t)‖L∞ . ε12−(N2−1/2)l〈t〉−1/2,

‖f±l (t)‖L∞ . ε12−(N2+1/2)l.

(9.3)

Using (8.52) and the symbol bounds (8.6) and (8.19), it is not hard to see that∣∣cι1ι2ι3(ξ, η, σ) · ϕ′k1(ξ − η)ϕ′k2(η − σ)ϕ′k3(σ)∣∣

. 23 max(k1,k2,k3)/22[med(k1,k2,k3)−min(k1,k2,k3)]/2.(9.4)

Using this one can decompose the integrals Iι1ι2ι3 into sums of the integrals Iι1ι2ι3k1,k2,k3, and then

estimate the terms corresponding to large frequencies, and the terms corresponding to smallfrequencies (relative to m), using only the bounds (9.3)-(9.4). As in [32, Section 6], we canthen reduce matters to proving the following:

Lemma 9.1. Assume that k ∈ [−80p0m, 20p0m], |ξ| ∈ [2k, 2k+1] ∩ Z, m ≥ 1, t1 ≤ t2 ∈[2m − 2, 2m+1] ∩ [0, T ′], and k1, k2, k3 are integers satisfying

k1, k2, k3 ∈ [−3m, 3m/N0 − 1000],

min(k1, k2, k3)/2 + 3med(k1, k2, k3)/2 ≥ −m(1 + 10p0).(9.5)

Then ∣∣∣ ∫ t2

t1

eiL(ξ,s)[I++−k1,k2,k3

(ξ, s) + c(ξ)f+k1

(ξ, s)f+k2

(ξ, s)f−k3(−ξ, s)s+ 1

]ds∣∣∣ . ε3

12−120p0m, (9.6)

and, for (ι1, ι2, ι3) ∈ (+ + +), (−−+), (−−−),∣∣∣ ∫ t2

t1

eiL(ξ,s)Iι1ι2ι3k1,k2,k3(ξ, s) ds

∣∣∣ . ε312−120p0m. (9.7)

Moreover ∣∣∣ ∫ t2

t1

eiL(ξ,s)e−is|ξ|3/2R′′≥4(ξ, s) ds

∣∣∣ . ε312−120p0m. (9.8)

The rest of this section is concerned with the proof of this lemma. We will often use thealternative formulas

Iι1ι2ι3k1,k2,k3(ξ, t) =

∫R×R

eitΦι1ι2ι3 (ξ,η,σ)c∗,ι1ι2ι3ξ;k1,k2,k3

(η, σ)f ι1k1(ξ+η)f ι2k2(ξ+σ)f ι3k3(−ξ−η−σ) dηdσ, (9.9)

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WATER WAVES WITH SURFACE TENSION 61

where

Φι1ι2ι3(ξ, x, y) := −Λ(ξ) + ι1Λ(ξ + x) + ι2Λ(ξ + y) + ι3Λ(ξ + x+ y),

c∗,ι1ι2ι3ξ;k1,k2,k3(x, y) := cι1ι2ι3(ξ,−x,−ξ − x− y) · ϕ′k1(ξ + x)ϕ′k2(ξ + y)ϕ′k3(ξ + x+ y).

(9.10)

These formulas follow from (9.1) via simple changes of variables.We now recall that the symbols cι1ι2ι3 are given by the explicit formulas (8.52), together

with the formulas (8.11) for mε1ε2 , and (8.4) for qε1ε2 . Using the symbol bounds (8.19) and theexplicit formulas (C.58) one can verify that the symbols c∗,ι1ι2ι3ξ;k1,k2,k3

satisfy the S∞ estimates∥∥F−1(c∗,ι1ι2ι3ξ;k1,k2,k3

)∥∥L1(R2)

. 23kmax/22(kmed−kmin)/2, (9.11)

and, with ψξ;l1,l2,l3(x, y) = ϕl1(x)ϕl2(y)ϕl3(2ξ + x+ y),∥∥F−1[(∂xc

∗,ι1ι2ι3ξ;k1,k2,k3

)(x, y) · ψξ;l1,l2,l3(x, y)]∥∥L1(R2)

. 23kmax/22(kmed−kmin)/22−min(k1,k3),∥∥F−1[(∂yc

∗,ι1ι2ι3ξ;k1,k2,k3

)(x, y) · ψξ;l1,l2,l3(x, y)]∥∥L1(R2)

. 23kmax/22(kmed−kmin)/22−min(k2,k3),(9.12)

for any ξ ∈ R and k1, k2, k3, l1, l2, l3 ∈ Z, where kmax := max(k1, k2, k3), kmed := med(k1, k2, k3),kmin := min(k1, k2, k3). In fact the contributions of the cubic symbols mι1ι2ι3 satisfy stronger

bounds, with a factor of |ξ|1/22kmax instead of 23kmax/22(kmed−kmin)/2, see (C.58).The cutoff factors ψξ;l1,l2,l3(x, y) are needed in order for the bounds (9.12) to hold, due to

the presence of factors such as |x| in the symbols. These cutoffs do not play an important rolein the proof of Lemma 9.1. Inserting these cutoffs only leads to an additional logarithmic lossin 〈t〉 in some of the estimates, which we can easily tolerate.

9.1. Proof of (9.6). We divide the proof of the bound (9.6) into several lemmas. Since in thissubsection we will only deal with interactions of the type + +−, for simplicity of notation wedenote Φ := Φ++− and c∗k := c∗,++−

ξ;k1,k2,k3.

Lemma 9.2. The bound (9.6) holds provided that (9.5) holds, and

max(|k − k1|, |k − k2|, |k − k3|

)≤ 20.

Proof. This is the case which gives the precise form of the correction. However, the proofis similar to the proof of Lemma 6.4 in [32]. The only difference is that in the present case

Λ(ξ) = |ξ|3/2 instead of |ξ|1/2. Therefore one has the expansion∣∣∣Λ(ξ)− Λ(ξ + η)− Λ(ξ + σ) + Λ(ξ + η + σ)− 3ησ

4|ξ|1/2∣∣∣ . 2−3k/2(|η|3 + |σ|3).

One can then follow the same argument in [32, Lemma 6.4] to obtain the desired bound.

Lemma 9.3. The bound (9.6) holds provided that (9.5) holds and, in addition,

max(|k − k1|, |k − k2|, |k − k3|) ≥ 21,

max(|k1 − k3|, |k2 − k3|) ≥ 5 and min(k1, k2, k3) ≥ − 49

100m.

(9.13)

Proof. In this case we will show the stronger bound

|I++−k1,k2,k3

(ξ, s)| . ε312−m2−200p0m. (9.14)

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62 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Without loss of generality, by symmetry we can assume that |k1−k3| ≥ 5 and k2 ≤ max(k1, k3)+5. Under the assumptions (9.13) we have

|(∂ηΦ)(ξ, η, σ)| = |Λ′(ξ + η)− Λ′(ξ + η + σ)| & 2kmax/2. (9.15)

Therefore we can integrate by parts in η in the integral expression (9.9) for I++−k1,k2,k3

. This gives

|I++−k1,k2,k3

(ξ, s)| . |K1(ξ, s)|+ |K2(ξ, s)|+ |K3(ξ, s)|+ |K4(ξ, s)|,

where

K1(ξ) :=

∫R×R

eisΦ(ξ,η,σ)m1(η, σ)c∗k(η, σ)(∂f+k1

)(ξ + η)f+k2

(ξ + σ)f−k3(−ξ − η − σ) dηdσ,

K2(ξ) :=

∫R×R

eisΦ(ξ,η,σ)m1(η, σ)c∗k(η, σ)f+k1

(ξ + η)f+k2

(ξ + σ)(∂f−k3)(−ξ − η − σ) dηdσ,

K3(ξ) :=

∫R×R

eisΦ(ξ,η,σ)(∂ηm1)(η, σ)c∗k(η, σ)f+k1

(ξ + η)f+k2

(ξ + σ)f−k3(−ξ − η − σ) dηdσ,

K4(ξ) :=

∫R×R

eisΦ(ξ,η,σ)m1(η, σ)(∂ηc∗k)(η, σ)f+

k1(ξ + η)f+

k2(ξ + σ)f−k3(−ξ − η − σ) dηdσ,

(9.16)

having denoted

m1(η, σ) :=1

s∂ηΦ(ξ, η, σ)ϕ′k1(ξ + η)ϕ′k3(ξ + η + σ). (9.17)

Observe that, under the restrictions (9.13), we have

‖m1(η, σ)‖S∞ . 2−m2−kmax/2. (9.18)

We can then estimate K1 using Lemma 2.1(ii), the estimate on c∗k in (9.11), the bounds (9.2)-

(9.3), and the last constraint in (9.13). More precisely, if k1 ≤ k3 (so that 2k3 ≈ 2kmax) thenwe estimate

|K1(ξ, s)| . ‖m1(η, σ)‖S∞‖c∗k(η, σ)‖S∞‖∂f

+k1

(s)‖L2‖eisΛf+

k2(s)‖

L∞‖f−k3(s)‖

L2

. 2−m2−k3/2 · 23k+3 2−kmin/2 · ε12−k1(1/2+p0)26mp0 · ε12−m/2 · ε126mp02−(N0−1)k+3

. ε312−3m/2212p0m2−kmin/22−k1(1/2+p0)

. ε31215p0m2−3m/22−kmin .

This suffices to prove (9.14) because of the last inequality in (9.13). On the other hand, ifk1 ≥ k3 (so that 2k1 ≈ 2kmax & 2k) then

|K1(ξ, s)| . ‖m1(η, σ)‖S∞‖c∗k(η, σ)‖S∞‖∂f

+k1

(s)‖L2‖eisΛf+

k2(s)‖

L∞‖f−k3(s)‖

L2

. 2−m2−k1/2 · 22k12−kmin/2 · ε12−k1−k+1 26mp0 · ε12−m/2 · ε126p0m

. ε312−3m/2214p0m2−kmin/2.

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WATER WAVES WITH SURFACE TENSION 63

This suffices to prove (9.14). The estimate for |K2(ξ, s)| is identical. The estimates for |K3(ξ, s)|and |K4(ξ, )| are also similar, because differentiation with respect to η is equivalent7 to multi-plication by a factor of (2−k1 + 2−k3), and one can use similar estimates as before

Lemma 9.4. The bound (9.6) holds provided that (9.5) holds and, in addition,

max(|k − k1|, |k − k2|, |k − k3|) ≥ 21,

max(|k1 − k3|, |k2 − k3|) ≥ 5 and min(k1, k2) ≤ −48m

100.

(9.19)

Proof. By symmetry we may assume that k2 = min(k1, k2). The main observation is that westill have the strong lower bound

|(∂ηΦ)(ξ, η, σ)| = |Λ′(ξ + η)− Λ′(ξ + η + σ)| & 2−kmax/22k.

This is easy to verify since |ξ| ≥ 2−80p0m, |ξ+σ| ≤ 2k2+1 ≤ 2−48m/100+1. We can then integrate

by parts in η, and estimate the resulting integrals as in Lemma 9.3, by placing fk2 in L2, andusing the restriction kmin + 3kmed ≥ −2m(1 + 10p0) in (9.5).

Lemma 9.5. The desired bound (9.6) holds provided that (9.5) holds and, in addition,

max(|k − k1|, |k − k2|, |k − k3|) ≥ 21,

max(|k1 − k3|, |k2 − k3|) ≥ 5 and min(k1, k2) ≥ −48m

100and k3 ≤ −

49m

100.

(9.20)

Proof. In this case we need to integrate by parts in time. Without loss of generality, we mayagain assume k2 = min(k1, k2). Therefore, also in view of (9.5), we have

k3 ≤ −49m/100 ≤ −48m/100 ≤ k2 ≤ k1, k1 ≥ k − 10 ≥ −80p0m− 10. (9.21)

Recall that

Φ(ξ, η, σ) = −Λ(ξ) + Λ(ξ + η) + Λ(ξ + σ)− Λ(ξ + η + σ).

For |ξ+ η| ≈ 2k1 , |ξ+ σ| ≈ |η| ≈ 2k2 , |ξ+ η+ σ| ≈ 2k3 , with k1, k2, k3 satisfying (9.21), one canuse (A.11) to show that

|Φ(ξ, η, σ)| ≥ | − Λ(ξ) + Λ(ξ + η) + Λ(η)| − 210|ξ + η + σ|2k2/2 & 2k22k/2. (9.22)

Thanks to this lower bound we can integrate by parts in s to obtain∣∣∣ ∫ t2

t1

eiL(ξ,s)I++−k1,k2,k3

(ξ, s) ds∣∣∣ . |N1(ξ, t1)|+ |N1(ξ, t2)|

+

∫ t2

t1

|N2(ξ, s)|+ |N3(ξ, s)|+ |N4(ξ, s)|+ |(∂sL)(ξ, s)||N1(ξ, s)| ds,(9.23)

7Some care is needed when the derivative hits c∗k, because of the slightly weaker bounds in (9.12). In thiscase we insert cutoff functions localizing the variables η, σ, 2ξ + η + σ to dyadic intervals and estimate in thesame way. The final bound is multiplied by Cm3, which is acceptable.

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64 ALEXANDRU D. IONESCU AND FABIO PUSATERI

where

N1(ξ) :=

∫R×R

eisΦ(ξ,η,σ) c∗k(η, σ)

iΦ(ξ, η, σ)f+k1

(ξ + η)f+k2

(ξ + σ)f−k3(−ξ − η − σ) dηdσ,

N2(ξ) :=

∫R×R

eisΦ(ξ,η,σ) c∗k(η, σ)

iΦ(ξ, η, σ)(∂sf

+k1

)(ξ + η)f+k2

(ξ + σ)f−k3(−ξ − η − σ) dηdσ,

N3(ξ) :=

∫R×R

eisΦ(ξ,η,σ) c∗k(η, σ)

iΦ(ξ, η, σ)f+k1

(ξ + η)(∂sf+k2

)(ξ + σ)f−k3(−ξ − η − σ) dηdσ,

N4(ξ) :=

∫R×R

eisΦ(ξ,η,σ) c∗k(η, σ)

iΦ(ξ, η, σ)f+k1

(ξ + η)f+k2

(ξ + σ)(∂sf−k3

)(−ξ − η − σ) dηdσ.

(9.24)

To estimate the first term in (9.23) we first notice that we have the pointwise bound∣∣∣ c∗k(η, σ)

Φ(ξ, η, σ)

∣∣∣ . 22k+1 2−k/22−k2/22−k3/2, (9.25)

see (9.22) and (9.11). Using also (9.2)-(9.3) we can obtain, for any s ∈ [t1, t2],

|N1(ξ, s)| . 22k+1 2−k/22−k2/22−k3/2‖f+k1

(s)‖L∞

2k2/2‖f+k2

(s)‖L22k3/2‖f−k3(s)‖

L2

. ε312k3(1/2−p0)2100p0m

. ε312−m/10.

Moreover, the definition of L in (8.57), and the a priori assumptions (9.2)-(9.3), show that∣∣(∂sL)(ξ, s)∣∣ . ε2

12−m.

Therefore

|N1(ξ, t1)|+ |N1(ξ, t2)|+∫ t2

t1

|(∂sL)(ξ, s)||N1(ξ, s)| ds . ε312−m/10. (9.26)

Using the equation for ∂tf in (8.23), and the estimate (8.29), we have∥∥(∂sf±l )(s)

∥∥L2 . ε

21 min(2(1/2−p0)l, 2−2l)2−m+6p0m. (9.27)

Using this L2 bound, and the pointwise bound (9.25) on the symbol, the term N2 can beestimated as follows:

|N2(ξ, s)| . 22k+1 2−k/22−k2/22−k3/2∥∥(∂sf

+k1

)(s)∥∥L2‖f+

k2(s)‖

L2‖f−k3(s)‖L22k3/2

. ε3122 max(k1,0)2−m+100p0m2−4k+2 2k3(1/2−p0)

. ε312−m2−m/10.

The integral |N3(ξ, s)| can be estimated in the same way. To deal with the last term in (9.24)we use again (9.25), (9.27), and the a priori bounds (9.2)-(9.3) to obtain:

|N4(ξ, s)| . 22 max(k1,0)2−k/22−k2/22−k3/2∥∥f+

k1(s)∥∥L2‖f+

k2(s)‖

L2‖(∂sf−k3)(s)‖L22k3/2

. ε312−m+100p0m2k3(1/2−p0)

. ε312−m2−m/10.

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WATER WAVES WITH SURFACE TENSION 65

We deduce that ∫ t2

t1

|N2(ξ, s)|+ |N3(ξ, s)|+ |N4(ξ, s)| ds . ε312−m/10,

and the lemma follows from (9.23) and (9.26).

Lemma 9.6. The bound (9.6) holds provided that (9.5) holds and, in addition,

max(|k − k1|, |k − k2|, |k − k3|) ≥ 21 and max(|k1 − k3|, |k2 − k3|) ≤ 4. (9.28)

Proof. In this case we have min(k1, k2, k3) ≥ k + 10, so that, in particular, |σ| ≈ 2k2 . Since allinput frequencies are comparable in view of the second assumption in (9.28), we can see that

|(∂ηΦ)(ξ, η, σ)| = |Λ′(ξ + η)− Λ′(ξ + η + σ)| & 2kmax/2,

which is the same lower bound as in (9.15). We can then integrate by parts in η similarly towhat was done before in Lemma 9.3. This gives

|I++−k1,k2,k3

(ξ, s)| . |K1(ξ, s)|+ |K2(ξ, s)|+ |K3(ξ, s)|+ |K4(ξ, s)|

where the term Kj , j = 1, . . . 4 are defined in (9.16)-(9.17), and the bound (9.17) is satisfied.Then, the same estimates that followed (9.18) show that |I++−

k1,k2,k3(ξ, s)| . ε312−m2−200p0m, which

suffices to prove the lemma.

9.2. Proof of (9.7). We divide the proof of the bound (9.7) into several lemmas. We onlyconsider in detail the case (ι1ι2ι3) = (− − +), since the cases (ι1ι2ι3) = (+ + +) or (− − −)

are very similar. In the rest of this subsection we let Φ := Φ−−+ and c∗k := c∗,−−+ξ;k1,k2,k3

.

Lemma 9.7. The bound (9.7) holds provided that (9.5) holds and, in addition,

max(|k1 − k3|, |k2 − k3|) ≥ 5 and min(k1, k2, k3) ≥ − 49

100m. (9.29)

Proof. This case is similar to the proof of Lemma 9.3. Without loss of generality, by symmetrywe can assume that |k1 − k3| ≥ 5 and k2 ≤ max(k1, k3) + 5. Under the assumptions (9.29) westill have the strong lower bound

|(∂ηΦ)(ξ, η, σ)| = |Λ′(ξ + η)− Λ′(ξ + η + σ)| & 2kmax/2.

The proof can then proceed exactly as in Lemma 9.3, using integration by parts in η.

Lemma 9.8. The bound (9.7) holds provided that (9.5) holds and, in addition,

max(|k1 − k3|, |k2 − k3|) ≥ 5 and med(k1, k2, k3) ≤ −48m/100. (9.30)

Proof. This is similar to the situation in Lemma 9.4. By symmetry we may assume thatk2 = min(k1, k2). The main observation is that in this case we must have |k1−k3| ≥ 5, becauseof the second assumption in (9.30) and k ≥ −80p0m. We then have the lower bound

|(∂ηΦ)(ξ, η, σ)| = |Λ′(ξ + η)− Λ′(ξ + η + σ)| & 2kmax/2.

Thus, we integrate by parts in η and estimate the resulting integrals as in Lemma 9.3, by

placing f+k2

in L2, and using the restriction kmin + 3kmed ≥ −2m(1 + 10p0), see (9.5).

Lemma 9.9. The bound (9.7) holds provided that (9.5) holds and, in addition,

max(|k1 − k3|, |k2 − k3|) ≥ 5 and kmed ≥ −48m

100and kmin ≤ −

49m

100.

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66 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Proof. This situation is similar to the one in Lemma 9.5. The main observation is that we havethe lower bound

|Φ(ξ, η, σ)| & 2min(kmed,k)2k/2,

so that we can integrate by parts in time and estimate the resulting integrals as in the proofof Lemma 9.5.

Lemma 9.10. The bound (9.7) holds provided that (9.5) holds and, in addition,

max(|k − k1|, |k − k2|, |k − k3|) ≥ 21 and max(|k1 − k3|, |k2 − k3|) ≤ 4.

Proof. This case is similar to that of Lemma 9.6. Observing that k + 10 ≤ min(k1, k2, k3), wesee that

|(∂ηΦ)(ξ, η, σ)| = |Λ′(ξ + η)− Λ′(ξ + η + σ)| & 2kmax/2.

We can then use again integration by parts in η to obtain the desired bound.

Lemma 9.11. The bound (9.7) holds provided that (9.5) holds and, in addition,

max(|k − k1|, |k − k2|, |k − k3|) ≤ 20. (9.31)

Proof. This is the main case where there is a substantial difference between the integrals I++−k1,k2,k3

and I−−+k1,k2,k3

. The main point is that the phase function Φ−−+ does not have any spacetime

resonances, i.e. there are no (η, σ) solutions of the equations

Φ−−+(ξ, η, σ) = (∂ηΦ−−+)(ξ, η, σ) = (∂σΦ−−+)(ξ, η, σ) = 0.

For any l, j ∈ Z satisfying l ≤ j define

ϕ(l)j :=

ϕj if j ≥ l + 1,

ϕ≤l if j = l.

Let l := k − 20 and decompose

I−−+k1,k2,k3

=∑

l1,l2∈[l,k+40]

Jl1,l2 ,

Jl1,l2(ξ, t) :=

∫R×R

eitΦ(ξ,η,σ)c∗k(η, σ)ϕ(l)l1

(η)ϕ(l)l2

(σ)f−k1(ξ + η)f−k2(ξ + σ)f+k3

(−ξ − η − σ) dηdσ.

The contributions of the integrals Jl1,l2 for (l1, l2) 6= (l, l) can be estimated by integration byparts either in η or in σ (depending on the relative sizes of l1 and l2), since the (η, σ) gradient

of the phase function Φ is bounded from below by c2k/2 in the support of these integrals.On the other hand, to estimate the contribution of the integral Jl,l we notice that

|Φ(ξ, η, σ)| & 23k/2

in the support of the integral, so that we can integrate by parts in s. This gives∣∣∣ ∫ t2

t1

eiL(ξ,s)Jl,l(ξ, s) ds∣∣∣ . |L4(ξ, t1)|+ |L4(ξ, t2)|

+

∫ t2

t1

|L1(ξ, s)|+ |L2(ξ, s)|+ |L3(ξ, s)|+ |(∂sL)(ξ, s)||L4(ξ, s)| ds,(9.32)

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WATER WAVES WITH SURFACE TENSION 67

where

L1(ξ) :=

∫R×R

eisΦ(ξ,η,σ)c∗k(η, σ)ϕ≤l(η)ϕ≤l(σ)

iΦ(ξ, η, σ)(∂sf

−k1

)(ξ + η)f−k2(ξ + σ)f+k3

(−ξ − η − σ) dηdσ,

L2(ξ) :=

∫R×R

eisΦ(ξ,η,σ)c∗k(η, σ)ϕ≤l(η)ϕ≤l(σ)

iΦ(ξ, η, σ)f−k1(ξ + η)(∂sf

−k2

)(ξ + σ)f+k3

(−ξ − η − σ) dηdσ,

L3(ξ) :=

∫R×R

eisΦ(ξ,η,σ)c∗k(η, σ)ϕ≤l(η)ϕ≤l(σ)

iΦ(ξ, η, σ)f−k1(ξ + η)f−k2(ξ + σ)(∂sf

+k3

)(−ξ − η − σ) dηdσ,

L4(ξ) :=

∫R×R

eisΦ(ξ,η,σ)c∗k(η, σ)ϕ≤l(η)ϕ≤l(σ)

iΦ(ξ, η, σ)f−k1(ξ + η)f−k2(ξ + σ)f+

k3(−ξ − η − σ) dηdσ.

To estimate the integrals L1, L2, L3, L4 we notice that using (9.11) and (9.31) we have∥∥∥c∗k(η, σ)ϕ≤l(η)ϕ≤l(σ)

iΦ(ξ, η, σ)

∥∥∥S∞. 23k+ .

Therefore, using Lemma 2.1(ii) and the a priori bounds (9.2)-(9.3) we see that

|L4(ξ, s)| .∥∥∥c∗k(η, σ)ϕ≤l(η)ϕ≤l(σ)

iΦ(ξ, η, σ)

∥∥∥S∞‖f−k1(s)‖

L2‖eisΛf−k2(s)‖L∞‖f+k3

(s)‖L2 . ε

312−m/4.

Using also (9.27) we obtain

|L1(ξ, s)| .∥∥∥c∗k(η, σ)ϕ≤l(η)ϕ≤l(σ)

iΦ(ξ, η, σ)

∥∥∥S∞‖(∂sf−k1)(s)‖

L2‖eisΛf−k2(s)‖L∞‖f+k3

(s)‖L2 . ε

312−5m/4.

The bounds on |L2(ξ, s)| and |L3(ξ, s)| are similar. Recalling also the bound |(∂sL)(ξ, s)| .ε2

12−m, see the definition (8.57), it follows that the right-hand side of (9.32) is dominated by

Cε312−m/10, which completes the proof of the lemma.

9.3. Proof of (9.8). We consider now the quartic remainder term R′′≥4 defined in (8.47). Our

main aim is to show the following L2 estimates:

Lemma 9.12. For any t ∈ [0, T ′] and l ∈ Z we have

‖PlR′′≥4(t)‖

L2 . ε41〈t〉−9/8+10p0 , ‖PlSR

′′≥4(t)‖

L2 . ε41〈t〉−1+20p0 . (9.33)

The desired conclusion (9.8) can then obtained using also the interpolation inequality inLemma 2.3; the precise argument is given after the proof of Lemma 9.12.

To prove (9.33) we first recall that from the a priori assumptions on u and Lemma 8.2 wehave the linear bounds

‖PlU(t)‖L2 + ‖Plv(t)‖L2 . ε1 min(2(1/2−p0)l, 2−(N0−1/2)l

)〈t〉p0 ,

‖PlU(t)‖L∞ + ‖Plv(t)‖L∞ . ε1 min(2l/10, 2−(N2−1/2)l

)〈t〉−1/2,

‖PlSU(t)‖L2 + ‖PlSv(t)‖L2 . ε1 min(2(1/2−p0)l, 2−(N1−1/2)l

)〈t〉4p0 ,

(9.34)

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68 ALEXANDRU D. IONESCU AND FABIO PUSATERI

and the quadratic bounds

‖Pl(U(t)− v(t))‖L2 . ε21 min

(2l/2, 2−(N2−1/2)l

)〈t〉−1/2+6p0 ,

‖Pl(U(t)− v(t))‖L∞ . ε21 min

(2l/2, 2−(N2−1/2)l

)〈t〉−3/4+2p0 ,

‖PlS(U(t)− v(t))‖L2 . ε21 min

(2l/2, 2−(N1−1/2)l

)〈t〉−1/4+6p0 .

(9.35)

Using these we now prove a few more nonlinear estimates.

Lemma 9.13. For any t ∈ [0, T ′] and l ∈ Z we have the quadratic-type bounds

‖PlQU (t)‖L2 + ‖PlQv(t)‖L2 . ε21 min

(2l/2, 2−(N0−3/2)l

)〈t〉−1/2+p0 ,

‖PlQU (t)‖L∞ + ‖PlQv(t)‖L∞ . ε21 min

(2l/2, 2−3l

)〈t〉−1,

‖PlSQU (t)‖L2 + ‖PlSQv(t)‖L2 . ε21 min

(2l/2, 2−l

)〈t〉−1/2+4p0 ,

(9.36)

and the cubic-type bounds

‖Pl(QU (t)−Qv(t))‖L2 . ε31〈t〉−7/8+6p0 min(2l/2, 2−3l),

‖Pl(QU (t)−Qv(t))‖L∞ . ε31〈t〉−5/4+2p0 min(2l/2, 2−3l),

‖PlS(QU (t)−Qv(t))‖L2 . ε31〈t〉−3/4+8p0 min(2l/2, 2−l).

(9.37)

Proof. To obtain (9.36) is suffices to recall the definition of QU and Qv, in (8.3) and (8.44),and use the symbol bounds (8.19),

‖qk,k1,k2ε1ε2 ‖S∞. 1X (k, k1, k2)2k2min(k,k1,k2)/2 (9.38)

the commutation identity (2.5)-(2.6), and the linear bounds (9.34) (see the proof of LemmaC.2 for a similar argument).

We now prove the inequalities (9.37). In view of the definitions (8.3) and (8.44) we have

QU −Qv =∑?

′[Qε1ε2(Uε1 , Uε2 − vε2) +Qε1ε2(Uε1 − vε1 , vε2)

]. (9.39)

Using Lemma 2.1(ii) with (9.38), the L∞ estimate in (9.34) and the first L2 estimate in(9.35), we get

‖PkQε1ε2(Uε1 , Uε2 − vε2)(t)‖L2 .∑

k1,k2∈Z‖qk,k1,k2ε1ε2 ‖

S∞‖P ′k1U(t)‖

L∞‖P ′k2(U − v)(t)‖

L2

. ε31〈t〉−1+6p0

∑k1,k2∈Z

1X (k, k1, k2)2k2min(k1,k2)/22−(N2−1/2)k+1 2−(N2−1/2)k+2

. ε31〈t〉−1+6p0 min(2k/2, 2−(N2−3/2)k).

The first bound in (9.37) follows, after estimating similarly the L2 norm of the second termin (9.39). The L∞ bound in (9.37) can also be obtained similarly using the identity (9.39) andthe L∞ bounds in (9.34)-(9.35). For the last bound in (9.37) we first use (9.39) and (2.5)-(2.6)(notice that the symbols qε1ε2 are homogeneous). Then we estimate the L2 norm in the sameway, placing SU and S(U − v) in L2 and U − v and U in L∞.

We are now ready to prove (9.33).

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WATER WAVES WITH SURFACE TENSION 69

Proof of Lemma 9.12. We examine the formula (8.47) for R′′≥4, and begin by looking at theterms in the second line:

Mε1ε2(Uε1 , (QU )ε2)−Mε1ε2(vε1 , (Qv)ε2)

= Mε1ε2(Uε1 − vε1 , (QU )ε2) +Mε1ε2(vε1 , (QU )ε2 − (Qv)ε2).(9.40)

We recall the bound (8.19),

‖mk,k1,k2ε1ε2 ‖

S∞. 2k/22−min(k1,k2)/21X (k, k1, k2), (9.41)

and remark that the difficulty in estimating the terms in (9.40) comes from the low frequencysingularity in this estimate. Using Lemma 2.1(ii), (9.35), and (9.36) we have∥∥PlMε1ε2(Uε1 − vε1 , (QU )ε2)(t)

∥∥L2 . I + II,

I :=∑

k1,k2∈Z, 2min(k1,k2)≤〈t〉−2

‖ml,k1,k2ε1ε2 ‖S∞2min(k1,k2)/2‖P ′k1(U − v)(t)‖

L2‖P ′k2QU (t)‖L2 ,

II :=∑

k1,k2∈Z, 2min(k1,k2)≥〈t〉−2

‖ml,k1,k2ε1ε2 ‖S∞‖P

′k1(U − v)(t)‖

L2‖P ′k2QU (t)‖L∞.

Then we estimate

I .∑

2min(k1,k2)≤〈t〉−2

ε21〈t〉−1/2+6p0ε2

1〈t〉−1/2+p02min(k1,k2)/22−3 max(k1,k2,0) . ε41〈t〉−3/2

and

II .∑

2min(k1,k2)≥〈t〉−2

ε21〈t〉−1/2+6p0ε2

1〈t〉−12−2 max(k1,k2,0) . ε41〈t〉−5/4.

Similarly, using (9.34) (only the L2 estimate in the first line) and (9.37) instead of (9.35)and (9.36), we estimate∥∥PlMε1ε2(vε1 , (QU )ε2 − (Qv)ε2)(t)

∥∥L2 . ε

41〈t〉−5/4+10p0 .

We have obtained, for any l ∈ Z and t ∈ [0, T ′],∥∥Pl[Mε1ε2(Uε1 , (QU )ε2)−Mε1ε2(vε1 , (Qv)ε2)](t)∥∥L2 . ε

41〈t〉−9/8. (9.42)

Using (9.40) and the commutation identity (2.5)-(2.6) we compute

S(Mε1ε2(Uε1 , (QU )ε2)−Mε1ε2(vε1 , (Qv)ε2)

)= Mε1ε2(S(Uε1 − vε1), (QU )ε2) +Mε1ε2(Uε1 − vε1 , (SQU )ε2) + Mε1ε2(Uε1 − vε1 , (QU )ε2)

+Mε1ε2(Svε1 , (QU −Qv)ε2) +Mε1ε2(vε1 , S(QU −Qv)ε2) + Mε1ε2(vε1 , (QU −Qv)ε2).

(9.43)

By homogeneity, the symbols mε1ε2 satisfy the same bounds as the symbols mε1ε2 . Therefore∥∥Mε1ε2(Uε1 , (QU )ε2)(t)− Mε1ε2(vε1 , (Qv)ε2)(t)∥∥L2 . ε

41〈t〉−9/8.

The terms containing the vector-field S can also be estimated in the same way, using thebounds (9.34)–(9.37) and (9.41). We always place the factor containing S in L2. For example,

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70 ALEXANDRU D. IONESCU AND FABIO PUSATERI

for the most difficult term, we can estimate∥∥PlMε1ε2(vε1 , S(QU −Qv)ε2)(t)∥∥L2 . IS + IIS ,

IS =∑

2min(k1,k2)≤〈t〉−1/2

2l/21X (l, k1, k2)‖P ′k1v(t)‖L2‖P ′k2S(QU −Qv)(t)

∥∥L2 ,

IIS :=∑

2min(k1,k2)≥〈t〉−1/2

2l/22−min(k1,k2)/21X (l, k1, k2)‖P ′k1v(t)‖L∞‖P ′k2S(QU −Qv)(t)

∥∥L2 .

Then we estimate

IS .∑

2min(k1,k2)≤〈t〉−1/2

ε1〈t〉p0ε31〈t〉−3/4+8p02min(k1,k2)(1/2−p0)2−max(k1,k2,0)/2 . ε4

1〈t〉−1+20p0

and

IIS .∑

2min(k1,k2)≥〈t〉−1/2

ε1〈t〉−1/2ε31〈t〉−3/4+8p02−min(k1,k2)/22−max(k1,k2,0)/2 . ε4

1〈t〉−1+20p0 .

The bound (9.33) for the terms in the second line of (8.47) follows, using also (9.42) and (9.43).The terms in the first line of (8.47) are similar to the terms in the second line, since the

bounds satisfied by the symbols of the operators Mε1ε2 are symmetric in k1 and k2. Next, welook at the terms in the third line of (8.47):

Mε1ε2ε3(Uε1 , Uε2 , Uε3)−Mε1ε2ε3(vε1 , vε2 , vε3) = Mε1ε2ε3

((U − v)ε1 , Uε2 , Uε3

)+Mε1ε2ε3

(vε1 , (U − v)ε2 , Uε3

)+Mε1ε2ε3

(vε1 , vε2 , (U − v)ε3

).

(9.44)

Recall that the symbols mε1ε2ε3 satisfy the strong (non-singular) bounds (8.6). Using (9.34)and (9.35), one can estimate∥∥Pl[Mε1ε2ε3(Uε1 , Uε2 , Uε3)−Mε1ε2ε3(vε1 , vε2 , vε3)

](t)∥∥L2 . ε

41〈t〉−9/8.

The analogue of (2.5) for symbols of three variables is

SM(f, g, h) = M(Sf, g, h) +M(f, Sg, h) +M(f, g, Sh) + M(f, g, h)

m(ξ, η, σ) = −(ξ∂ξ + η∂η + σ∂σ)m(ξ, η, σ).(9.45)

By homogeneity, the symbols mε1ε2ε3 satisfy the same bounds (8.6). Then, applying S to theidentity (9.44) above, and using (9.34) and (9.35), we can obtain∥∥PlS[Mε1ε2ε3(Uε1 , Uε2 , Uε3)−Mε1ε2ε3(vε1 , vε2 , vε3)

](t)∥∥L2 . ε

41〈t〉−9/8.

Since the term R≥4 in (8.47) already satisfies the desired bounds, see (8.8), in view of (8.9)it remains to estimate operators of the form

Mε1ε2(Fε1 , Uε2) and Mε1ε2(Uε1 , Fε2), F ∈ |∂x|1/2O3,−1. (9.46)

The O3,α notation in (C.3) implies that we have the following estimate

‖PlF (t)‖L2 . ε31 min

(2l/2, 2−(N0−2)l

)〈t〉−1+p0 ,

‖PlSF (t)‖L2 . ε31 min

(2l/2, 2−(N1−2)l

)〈t〉−1+4p0 .

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WATER WAVES WITH SURFACE TENSION 71

Using these estimates, Lemma 2.1(ii), the symbol bound (9.41), and (9.34), we can bound∥∥PlMε1ε2(Fε1 , Uε2)(t)∥∥L2 .

∑k1,k2∈Z, 2min(k1,k2)≤〈t〉−1/2

2max(k1,k2)/2‖P ′k1F (t)‖L2‖P ′k2U(t)‖

L2

+∑

k1,k2∈Z, 2min(k1,k2)≥〈t〉−1/2

2max(k1,k2)/22−min(k1,k2)/2‖P ′k1F (t)‖L2‖P ′k2U(t)‖

L∞

. ε41〈t〉−9/8.

The same estimate also holds for PlMε1ε2(Fε1 , Uε2). Similarly, we have∥∥PlMε1ε2(SFε1 , Uε2)(t)∥∥L2 .

∑k1,k2∈Z

2max(k1,k2)/2‖P ′k1SF (t)‖L2‖P ′k2U(t)‖

L2 . ε41〈t〉−1+10p0 ,

∥∥PlMε1ε2(Fε1 , SUε2)(t)∥∥L2 .

∑k1,k2∈Z

2max(k1,k2)/2‖P ′k1F (t)‖L2‖P ′k2SU(t)‖

L2 . ε41〈t〉−1+10p0 .

The estimates for the terms Mε1ε2(Uε1 , Fε2) are similar. This completes the proof of (9.33).

We can now complete the proof of the estimate (9.8).

Proof of (9.8). Assume that k ∈ [−80p0m, 20p0m], |ξ0| ∈ [2k, 2k+1], m ≥ 1, t1 ≤ t2 ∈ [2m −2, 2m+1] ∩ [0, T ′]. We would like to prove that∣∣∣ϕk(ξ0)

∫ t2

t1

eiL(ξ0,s)e−is|ξ0|3/2R′′≥4(ξ0, s) ds

∣∣∣ . ε312−200p0m. (9.47)

Let

F (ξ) := ϕk(ξ)

∫ t2

t1

eiL(ξ0,s)e−is|ξ|3/2R′′≥4(ξ, s) ds. (9.48)

In view of Lemma 2.3, it suffices to prove that

2−k‖F‖L2

[2k‖∂F‖L2 + ‖F‖L2

]. ε612−400p0m.

Since ‖F‖L2 . ε412−m/8+10p0m, see the first inequality in (9.33), it suffices to prove that

2k‖∂F‖L2 . ε412k2(1/8−500p0)m. (9.49)

To prove (9.49) we write

|ξ∂ξF (ξ)| ≤ |F1(ξ)|+ |F2(ξ)|+ |F3(ξ)|,where

F1(ξ) := ξ(∂ξϕk)(ξ)

∫ t2

t1

eiL(ξ0,s)[e−is|ξ|

3/2R′′≥4(ξ, s)]ds,

F2(ξ) := ϕk(ξ)

∫ t2

t1

eiL(ξ0,s)[ξ∂ξ − (3/2)s∂s

][e−is|ξ|

3/2R′′≥4(ξ, s)]ds,

F3(ξ) :=3

2ϕk(ξ)

∫ t2

t1

eiL(ξ0,s)s∂s[e−is|ξ|

3/2R′′≥4(ξ, s)]ds.

Using (9.33) and the commutation identity[[ξ∂ξ − (3/2)s∂s

], e−is|ξ|

3/2]= 0, we have

‖F1‖L2 + ‖F2‖L2 . ε41230p0m.

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72 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Moreover, using integration by parts in s and the bound∣∣∂s[eiL(ξ0,s)

]∣∣ . 2−m, see the definition

(8.57), we can also estimate ‖F3‖L2 . ε41230p0m. The desired bound (9.49) follows.

10. Modified scattering

In this section we provide a precise description of the asymptotic behavior of solutions. Ourmain result is the following:

Theorem 10.1 (Modified Scattering). Assume that N0, N1, N2, p0, p1 are as in Theorem 1.1,and (h, φ) is the global solution of the system (1.6). Let

U = |∂x|φ− i|∂x|1/2φ ∈ C([0,∞) : HN0,p1−1/2).

(i) (Scattering in the Fourier space) Let

L′(ξ, t) =|ξ|2

24π

∫ t

0|U(ξ, s)|

2 1

s+ 1ds, (10.1)

compare with (8.57). Then there is w∞ with ‖(1 + |ξ|N2)w∞(ξ)‖L2ξ. ε0 such that

eiL′(ξ,t)e−it|ξ|

3/2U(ξ, t) converges to w∞(ξ) as t→∞ ,

more precisely

(1 + t)p0/2∥∥(|ξ|−1/4 + |ξ|N2)[eiL

′(ξ,t)e−it|ξ|3/2U(ξ, t)− w∞(ξ)]

∥∥L2ξ. ε0. (10.2)

(ii) (Scattering in the physical space) There exists a unique asymptotic profile f∞, with ‖(|ξ|−1/10+|ξ|4)f∞‖L∞ξ . ε0, such that for all t ≥ 1∥∥∥U(x, t)− e−it(4/27)|x/t|3

√1 + t

f∞(x/t)e−id0|x/t|3|f∞(x/t)|2 log(1+t)

∥∥∥L∞x. ε0〈t〉−1/2−p0/2 , (10.3)

where d0 = 1/54.

We remark that the first statement of modified scattering, in the Fourier space, is strongerbecause of the stronger norm of convergence. One can, in fact, make this convergence evenstronger, by changing the norm in (10.2) to an L∞ξ based norm.

Modified scattering in the physical space follows by an argument similar to the one used byHayashi–Naumkin [26] (see also [33]).

To prove Theorem 10.1 we need a more precise scale invariant linear dispersive estimate:

Lemma 10.2. (i) For any t ∈ R \ 0, k ∈ Z, and f ∈ L2(R) we have

‖eitΛPkf‖L∞ . |t|−1/22k/4‖f‖L∞ + |t|−3/52−2k/5[‖f‖L2 + 2k‖∂f‖L2

](10.4)

and

‖eitΛPkf‖L∞ . |t|−1/22k/4‖f‖L1 . (10.5)

(ii) Moreover, if ξ0 := −4x2/(9t2) sgn(x/t) and f is continuous then∣∣∣eitΛf(x)−√

2i

3πte−it(4/27)|x/t|3 |ξ0|1/4f(ξ0)

∣∣∣ . |t|−3/5(‖|ξ|−2/5f‖L2 + ‖|ξ|3/5∂f‖L2

). (10.6)

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WATER WAVES WITH SURFACE TENSION 73

Proof. (i) This is similar to the proof of Lemma 2.3 in [31]. The bound (10.5) is a standarddispersive estimate. Also, (10.4) is a consequence of (10.6), so we focus on this last bound.

We start from the identity

eitΛf(x) =1

∫Rei(t|ξ|

3/2+xξ)f(ξ) dξ.

We would like to use the standard stationary phase approximate formula∫ReiΦ(x)Ψ(x) dx ≈ eiΦ(x0)Ψ(x0)

√2π

−iΦ′′(x0)+ error, (10.7)

in a neighborhood of a stationary point x0. To justify this we make the change of variables

ξ = bη, b := −4x2

9t2sgn(x/t), t′ := t · 4|x|3

9|t|3.

and notice that

t|ξ|3/2 + xξ = t′(2|η|3/2/3− η).

Therefore, letting fb(η) := f(bη), we have

eitΛf(x) =1

4x2

9t2

∫Reit′(2|η|3/2/3−η)fb(η) dη. (10.8)

Notice that

‖|η|−2/5fb(η)‖L2η

+ ‖|η|3/5(∂fb)(η)‖L2η

= |b|−1/10(‖|ξ|−2/5f(ξ)‖L2

ξ+ ‖|ξ|3/5(∂f)(ξ)‖L2

ξ

). (10.9)

For (10.6) it suffices to prove that∣∣∣ ∫Reit′(2|η|3/2/3−η)fb(η) dη − e−it′/3fb(1)

√4πi

t′

∣∣∣ . |t′|−3/5(‖|η|−2/5fb‖L2 + ‖|η|3/5∂fb‖L2

).

(10.10)

Let g := fb and gk := g · ϕk, k ∈ Z. Let

ak := 2−2k/5‖gk‖L2 + 23k/5‖∂gk‖L2 .

Let

Ψ(η) := 2|η|3/2/3− η, Ψ′(η) = η|η|−1/2 − 1, Ψ′′(η) = |η|−1/2/2. (10.11)

Clearly, for any k ∈ Z, ∣∣∣ ∫Reit′Ψ(η)gk(η) dη

∣∣∣ . ‖gk‖L1 . 29k/10ak.

Moreover, if |k| ≥ 3 then we can integrate by parts in η to estimate∣∣∣ ∫Reit′Ψ(η)gk(η) dη

∣∣∣ . ‖gk‖L1

|t′|23k/2+‖∂gk‖L1

|t′|2k/2.ak2−3k/5

|t′|.

These last two inequalities show that∑|k|≥3

∣∣∣ ∫Reit′Ψ(η)gk(η) dη

∣∣∣ . |t′|−3/5 supk∈Z

ak.

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74 ALEXANDRU D. IONESCU AND FABIO PUSATERI

One can also estimate, in the same way, the contribution of the negative frequencies of thefunctions gk, |k| ≤ 2. For (10.10) it remains to show that∣∣∣ ∫

Reit′Ψ(η)g(η) dη − e−it′/3g(1)

√4πi

t′

∣∣∣ . |t′|−3/5(‖g‖L2 + ‖∂g‖L2

). (10.12)

for any continuous function g supported in [1/10, 10].In proving (10.12) we may assume that |t′| is large, say |t′| ≥ 210. Let χ(η) = ϕ≤−10(η − 1)

denote a cutoff function supported around 1. For (10.12) it suffices to prove that∣∣∣ ∫Reit′Ψ(η)[g(η)− g(1)χ1(η)] dη

∣∣∣ . |t′|−3/5(‖g‖L2 + ‖∂g‖L2

)(10.13)

and, using also Lemma 2.3,∣∣∣ ∫Reit′Ψ(η)χ1(η) dη − e−it′/3

√4πi

t′

∣∣∣ . |t′|−3/5. (10.14)

The bound (10.14) follows easily by stationary phase and (10.11) (compare with (10.7)). Toprove (10.13) we define

Jl :=

∫Reit′Ψ(η)ϕl(η − 1)[g(η)− g(1)χ1(η)] dη,

for l ≤ 10. By linearity we may assume that ‖g‖L2 + ‖∂g‖L2 = 1. Notice that

if |η − 1| . 2l then |g(η)− g(1)χ1(η)| . |η − 1|1/2 . 2l/2. (10.15)

Therefore |Jl| . 23l/2. Moreover, if 2l ≥ |t′|−2/5 then we integrate by parts in η and estimate

|Jl| .∫R

∣∣∣ ddη

ϕl(η − 1)[g(η)− g(1)χ1(η)]

|t′|Ψ′(η)

∣∣∣ dη.∫|η−1|≤2l+1

2−l|g(η)− g(1)χ1(η)|+ |g′(η)|+ 1

2l|t′|dη.

Using (10.15) it follows that |Jl| . |t′|−12−l/2. The desired bound (10.13) follows, whichcompletes the proof.

We can complete now the proof of Theorem 10.1:

Proof of Theorem 10.1. (i) Recall the formulas (8.57),

L(ξ, t) =|ξ|2

24π

∫ t

0|v(ξ, s)|2 1

s+ 1ds, g(ξ, t) = eiL(ξ,t)e−it|ξ|

3/2v(ξ, t).

Assume 1 ≤ t1 ≤ t2, let H(t) := eiL′(ξ,t)e−it|ξ|

3/2U(ξ, t), and estimate∣∣[H(ξ, t2)−H(ξ, t1)]− [g(ξ, t2)− g(ξ, t1)]ei(L

′(ξ,t1)−L(ξ,t1))∣∣

=∣∣[ei(L′(ξ,t2)−L′(ξ,t1))e−it2|ξ|

3/2U(ξ, t2)− e−it1|ξ|3/2U(ξ, t1)]

− [ei(L(ξ,t2)−L(ξ,t1))e−it2|ξ|3/2v(ξ, t2)− e−it1|ξ|3/2 v(ξ, t1)]

∣∣. |U(ξ, t2)− v(ξ, t2)|+ |U(ξ, t1)− v(ξ, t1)|+ |v(ξ, t2)||(L′(ξ, t2)− L′(ξ, t1))− (L(ξ, t2)− L(ξ, t1))|.

(10.16)

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WATER WAVES WITH SURFACE TENSION 75

It follows from Lemma 8.6 that (|ξ|1/10 + |ξ|N2+1/2)|v(ξ, t)| . ε0 uniformly in t. Therefore

|v(ξ, t2)||(L′(ξ, t2)− L′(ξ, t1))− (L(ξ, t2)− L(ξ, t1))| .∫ t2

t1

|v(ξ, s)− U(ξ, s)| 1

s+ 1ds.

Using (8.14) and (10.16) it follows that∥∥(|ξ|−1/4 + |ξ|N2)[H(ξ, t2)−H(ξ, t1)]− [g(ξ, t2)− g(ξ, t1)]ei(L′(ξ,t1)−L(ξ,t1))

∥∥L2 . ε0〈t1〉−1/6.

Using now Lemma 8.6 and the bound (8.25) it follows that∥∥(|ξ|−1/4 + |ξ|N2)[H(ξ, t2)−H(ξ, t1)]∥∥L2 . ε0〈t1〉−2p0/3,

and the desired conclusion (10.2) follows.(ii) In view of (9.35),

‖U(t)− v(t)‖L∞ . ε0(1 + t)−2/3. (10.17)

Moreover, since v(ξ, t) = f(ξ, t)eit|ξ|3/2

= eit|ξ|3/2e−iL(ξ,t)g(ξ, t), and using also (8.25)–(8.26)

and (10.6), it follows that, for any x ∈ R and t ≥ 1,∣∣∣v(x, t)−√

2i

3πte−it(4/27)|x/t|3e−iL(ξ0,t)|ξ0|1/4g(ξ0, t)

∣∣∣ . 〈t〉−1/2−10p0 , (10.18)

where ξ0 := −4x2/(9t2) sgn(x/t).Let

g∞(ξ) := limt→∞|ξ|1/4g(ξ, t), (10.19)

where the limit exists due to Lemma 8.6; more precisely for any t ≥ 1∥∥∥g∞(ξ)− |ξ|1/4g(ξ, t)∥∥∥L∞ξ

. ε0(1 + t)−p0 , ‖|ξ|1/4g(ξ, t)‖L∞ξ . ε0. (10.20)

Then, for any ξ ∈ R,

L(ξ, t)− |ξ|3/2

24π|g∞(ξ)|2 ln(t+ 1) =

|ξ|3/2

24π

∫ t

0

[|ξ|1/2|g(ξ, t)|2 − |g∞(ξ)|2

] 1

s+ 1ds

Using (10.4) it follows that there is a real-valued function A∞ such that∥∥∥L(ξ, t)− |ξ|3/2

24π|g∞(ξ)|2 ln(t+ 1)−A∞(ξ)

∥∥∥L∞ξ

. ε0(1 + t)−2p0/3. (10.21)

Using (10.18), (10.20), and (10.21), it follows that∣∣∣v(x, t)−√

2i

3πte−it(4/27)|x/t|3ei[

|ξ0|3/2

24π|g∞(ξ0)|2 ln(t+1)+A∞(ξ0)]g∞(ξ0)

∣∣∣ . 〈t〉−1/2−2p0/3.

Then we define

f∞(y) :=

√2i

3πg∞(ξ(y))eiA∞(ξ(y)), ξ(y) := −4

9y2sgn(y).

The desired conclusion (10.3) follows using also (10.17).

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76 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Appendix A. Analysis of symbols

A.1. Notation. Recall the definition of the class of symbols

S∞ := m : Rd → C : m continuous and ‖m‖S∞ := ‖F−1(m)‖L1 <∞, (A.1)

and the notation

mk,k1,k2(ξ, η) := m(ξ, η)ϕk(ξ)ϕk1(ξ − η)ϕk2(η). (A.2)

Recall that X = (k, k1, k2) ∈ Z3 : max(k, k1, k2)−med(k, k1, k2) ≤ 6. Recall also the notation

m(ξ, η) = O(f(|ξ|, |ξ − η|, |η|)

)⇐⇒ ‖mk,k1,k2(ξ, η)‖S∞ . f(2k, 2k1 , 2k2)1X (k, k1, k2). (A.3)

A.2. Quadratic symbols. In the following lemma we collect several estimates on the symbolsa±± that are used throughout the paper.

Lemma A.1. Let ε1, ε2 ∈ +,−, and aε1ε2 be the symbols in (3.31)–(3.34), then

‖ak,k1,k2ε1ε2 ‖S∞. 23k1/21X (k, k1, k2)1[6,∞)(k2 − k1)

+ 2k1/22k1(−∞,1](k)1X (k, k1, k2)1[6,∞)(k2 − k1).(A.4)

Moreover, for ε1 ∈ +,−, the following bounds holds

‖ak,k1,k2ε1+ ‖S∞. 23k1/21X (k, k1, k2)1[6,∞)(k2 − k1), (A.5)

‖ak,k1,k2ε1− ‖S∞.(23k1/21[2,∞)(k) + 2k1/22k1(−∞,1](k)

)1X (k, k1, k2)1[6,∞)(k2 − k1). (A.6)

As a consequence, we have

∥∥∥ ak,k1,k2ε1ε2 (ξ, η)

|ξ|3/2 − ε1|ξ − η|3/2 − |η|3/2∥∥∥S∞. 2k1/22−k/21X (k, k1, k2)1[6,∞)(k2 − k1),

∥∥∥ ak,k1,k2ε1− (ξ, η)

|ξ|3/2 − ε1|ξ − η|3/2 + |η|3/2∥∥∥S∞.(23k1/22−3k/21[2,∞)(k) + 2k1/22−k/21(−∞,1](k)

)× 1X (k, k1, k2)1[6,∞)(k2 − k1).

(A.7)

Furthermore, for ε1 ∈ +,− let us define the symbol

αε1+(ξ, η, ρ) := aε1+(ξ, ρ)− aε1+(ξ + η − ρ, η). (A.8)

Then we have the following: if k3 ≤ k1 − 4, k4 ≤ k2 − 4, and k1 ≥ 8, then

‖αε1+(ξ, η, ρ) · ϕk1(ξ)ϕk2(η)ϕk3(ρ− ξ)ϕk4(ρ− η)‖S∞ . (25k3/2 + 25k4/2)(2k1 + 2k2)−1. (A.9)

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WATER WAVES WITH SURFACE TENSION 77

Proof. We begin by recalling that the cutoff χ, see (1.18), is supported on a region where23|ξ − η| ≤ |η|. Using integration by parts, one can verify that

ξ(ξ − η)

|ξ − η|1/2( |ξ||η|− 1)χ(ξ − η, η) = O

(|ξ − η|3/21[28,∞)(|η|/|ξ − η|)

),

η(ξ − η)

|ξ − η|1/2(

1− |ξ|1/2

|η|1/2)χ(ξ − η, η) = O

(|ξ − η|3/21[28,∞)(|η|/|ξ − η|)

),

|ξ − η|2|ξ|1/2

|η|χ(ξ − η, η) = O

(|ξ − η|2|η|−1/21[28,∞)(|η|/|ξ − η|)

),

|ξ|2 − |ξ|1/2|η|3/2

|η||ξ − η|1/2χ(ξ − η, η) = O

(|ξ − η|3/21[28,∞)(|η|/|ξ − η|)

)|ξ|2 + |ξ|1/2|η|3/2

|η||ξ − η|1/2ϕ≤0(η)χ(ξ − η, η) = O

(|ξ||ξ − η|1/21[28,∞)(|η|/|ξ − η|)1(0,23](|ξ|)

),

(A.10)

where we are using the notation (A.1)-(A.3). Since the bound for |ξ − η|3/2 is obvious, using(A.10), and inspecting the formulas (3.31)-(3.34), one immediately obtains (A.4). The bound(A.5) follows from the first four identities in (A.10). (A.6) follows directly from (A.4).

To prove (A.7) we notice first that

(a+ b)3/2 − b3/2 − a3/2 ∈ [ab1/2/4, 4ab1/2] if 0 ≤ a ≤ b. (A.11)

Therefore, using standard integration by parts, we see that∥∥∥ ϕk(ξ)ϕk1(ξ − η)ϕk2(η)

|ξ|3/2 − |ξ − η|3/2 − |η|3/2∥∥∥S∞.

1

2min(k1,k2)2max(k,k2)/2, (A.12)

for all k, k1, k2 ∈ Z. In particular, whenever k ≥ k1 + 3, we have∥∥∥ ϕk(ξ)ϕk1(ξ − η)ϕk2(η)

|ξ|3/2 − ε1|ξ − η|3/2 − |η|3/2∥∥∥S∞.

1

2k12max(k,k2)/2, (A.13)∥∥∥ ϕk(ξ)ϕk1(ξ − η)ϕk2(η)

|ξ|3/2 − ε1|ξ − η|3/2 + |η|3/2∥∥∥S∞.

1

23 max(k,k2)/2. (A.14)

Thus, (A.5) and (A.13) give the first inequality in (A.7). The second inequality in (A.7) is aconsequence of (A.6) and (A.14).

To prove (A.9) we write down the four components of the symbols,

a1(ξ, η) :=ξ(ξ − η)

|ξ − η|1/2( |ξ||η|− 1), a2(ξ, η) := |ξ − η|3/2,

a3(ξ, η) :=η(ξ − η)

|ξ − η|1/2(

1− |ξ|1/2

|η|1/2), a4(ξ, η) :=

|ξ − η|2|ξ|1/2

|η|.

(A.15)

Notice that the last component in the formulas (3.31) and (3.33) has been disregarded, sincewe are only interested in the case k1 ≥ 5 in (A.9). It suffices to prove that

‖αj(ξ, η, ρ) · ϕk1(ξ)ϕk2(η)ϕk3(ρ− ξ)ϕk4(ρ− η)‖S∞ . (25k3/2 + 25k4/2)(2k1 + 2k2)−1,

for αj(ξ, η, ρ) := aj(ξ, ρ)− aj(ξ + η − ρ, η), j = 1, . . . , 4,(A.16)

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78 ALEXANDRU D. IONESCU AND FABIO PUSATERI

whenever k3 ≤ k1−4 and k4 ≤ k2−4. By Taylor expansion, one easily sees that for 4|ξ−η| ≤ |η|

a1(ξ, η) = |ξ − η|3/2 +O(|ξ − η|5/2|η|−1

), a2(ξ, η) = |ξ − η|3/2

a3(ξ, η) = −1

2|ξ − η|3/2 +O

(|ξ − η|5/2|η|−1

), a4(ξ, η) = |ξ − η|2|η|−1/2 +O

(|ξ − η|3|η|−3/2

).

The desired conclusions in (A.16) follow.

We consider now the symbols b±±. Recall that χ(x, y) = 1− χ(x, y)− χ(y, x).

Lemma A.2. With bε1ε2 as (3.35)–(3.38) and (3.39), we have

‖bk,k1,k2ε1ε2 ‖S∞. 23k/21X (k, k1, k2)1[−15,15](k1 − k2), (A.17)

and ∥∥∥ bk,k1,k2ε1ε2

|ξ|3/2 − ε1|ξ − η|3/2 − ε2|η|3/2∥∥∥S∞. 2k/22−k1/21X (k, k1, k2)1[−15,15](k1 − k2), (A.18)

Proof. Inspecting the formula (3.39) it is easy to see that

m2(ξ, η) = O(|ξ||ξ − η|1[2−13,213](|ξ − η|/|η|)

),

and therefore

|ξ|m2(ξ, η)

|ξ − η|1/2|η|= O

(|ξ|3/21[2−13,213](|ξ − η|/|η|)

),

which is consistent with the bound in (A.17). Similarly

|ξ|1/2q2(ξ, η)

|ξ − η|1/2|η|1/2= O

(|ξ|3/21[2−13,213](|ξ − η|/|η|)

).

The desired conclusion (A.17) follows from these bounds and the formulas (3.35)-(3.38). Thebounds in (A.18) follow from (A.17) and (A.12).

Appendix B. The Dirichlet-Neumann operator

Recall the spaces C0, HN,b, WN,b, and WN defined in (1.8) and (2.2). Assume in this section

that h ∈ C([0, T ] : C0 ∩ HN0+1,1/2+p1

)satisfies the bounds∥∥h(t)∥∥HN0+1,1/2+p1,

. ε1〈t〉p0 ,∥∥h(t)∥∥WN2+1,9/10 . ε1〈t〉−1/2,∥∥Sh(t)

∥∥HN1+1,1/2+p1

. ε1〈t〉4p0 ,(B.1)

for any t ∈ [0, T ], which follow from the bootstrap assumption (2.25). For α ∈ [−1, 1] let

Eαw,T :=f ∈ C([0, T ] : HN0+α,p1) : ‖f‖Eαw,T := sup

t∈[0,T ]‖f(t)‖Eαw

, (B.2)

where

‖f(t)‖Eαw : = 〈t〉−p0‖f(t)‖HN0+α,p1 + 〈t〉1/2∥∥f(t)

∥∥WN2+α,2/5

+ 〈t〉−4p0‖Sf(t)‖HN1+α,p1 . (B.3)

Let G(h) denote the Dirichlet-Neumann operator defined in (1.5). Our main result in thissection is the following paralinearization of the operator G(h):

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WATER WAVES WITH SURFACE TENSION 79

Proposition B.1. Assume φ ∈ E1/2w,T and h satisfies (B.1), and define

B :=G(h)φ+ hxφx

1 + h2x

, V :=φx − hxG(h)φ

1 + h2x

. (B.4)

Then, for any f ∈ φx, G(h)φ and t ∈ [0, T ],

〈t〉−p0‖f‖HN0−1/2,−1/10 + 〈t〉1/2‖f‖WN2−1/2,−1/10 + 〈t〉−4p0‖Sf‖HN1−1/2,−1/10 . ‖φ‖E1/2w,T

. (B.5)

Moreover, we have the decomposition

G(h)φ = |∂x|φ− |∂x|TBh− ∂xTV h+G2(h, φ) +G≥3, (B.6)

where

G2(h, φ)(ξ) =1

∫Rh(η)φ(ξ − η)

[1− χ(ξ − η, η)

][ξ(ξ − η)− |ξ||ξ − η|

]dη, (B.7)

and, for any t ∈ [0, T ],

〈t〉1−p0‖G≥3(t)‖HN0+1 + 〈t〉11/10‖G≥3‖WN2+1 + 〈t〉1−4p0‖SG≥3(t)‖HN1+1 . ε21‖φ‖E1/2w,T

. (B.8)

The rest of this section is concerned with the proof of this proposition. We will need severalintermediate results. To estimate products we often use the following simple general lemma:

Lemma B.2. Assume a0, a2 ∈ [1/100, 100], A0, A2 ∈ (0,∞), and f, g ∈ L2(R) satisfy

A−10

(‖f‖Ha0 + ‖g‖Ha0

)+A−1

2

(‖f‖

Wa2+ ‖g‖

Wa2

)≤ 1. (B.9)

Then

A−10 ‖fg‖Ha0 +A−1

2 ‖fg‖Wa2. A2. (B.10)

Proof. Clearly,

‖fg‖L∞ . A22, ‖fg‖L2 . A0A2.

Moreover, for any k ≥ 0 and p ∈ 2,∞

‖Pk(fg)‖Lp . A2

∑k′≥k−4

[‖Pk′f‖Lp + ‖Pk′g‖Lp

].

The desired estimate follows.

B.1. The perturbed Hilbert transform and proof of Proposition B.1. With h as in(B.1) let

γ(x) := x+ ih(x) and Ω := x+ iy ∈ C : y ≤ h(x). (B.11)

Let D := ‖h‖L∞ . The finiteness of the (large) constant D is used to justify the convergence ofintegrals and some identities, but D itself does not appear in the main quantitative bounds.

For any f ∈ L2(R) we define the perturbed Hilbert transform

(Hγf)(α) :=1

πip.v.

∫R

f(β)γ′(β)

γ(α)− γ(β)dβ := lim

ε→0(Hεγf)(α),

(Hεγf)(α) :=1

πi

∫|β−α|≥ε

f(β)γ′(β)

γ(α)− γ(β)dβ.

(B.12)

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80 ALEXANDRU D. IONESCU AND FABIO PUSATERI

For any z ∈ Ω and f ∈ L2(R) we define

Ff (z) :=1

2πi

∫R

f(β)γ′(β)

z − γ(β)dβ. (B.13)

Clearly, Ff is an analytic function in Ω. For ε > 0 let

(T εγ f)(α) := Ff (γ(α)− iε) =1

2πi

∫R

f(β)γ′(β)

γ(α)− iε− γ(β)dβ. (B.14)

By comparing with the unperturbed case h = 0, it is easy to verify that, for any p ∈ (1,∞),∥∥ supε∈(0,∞)

|(Hεγf)(α)|∥∥Lpα

+∥∥ supε∈(0,∞)

|(T εγ f)(α)|∥∥Lpα.D,p ‖f‖Lp , (B.15)

and

limε→0Hεγf = Hγf, lim

ε→0T εγ f =

1

2(I +Hγ)f (B.16)

in Lp for any f ∈ Lp(R), p ∈ (1,∞).The perturbed Hilbert transform can be used to derive explicit formulas for G(h)φ (see

Lemma B.6). For this we will need a technical lemma:

Lemma B.3. Given h as in (B.1) and n ∈ Z+, we define the real operators

(Rnf)(α) :=1

π

∫R

h(α)− h(β)− h′(β)(α− β)

α− β

(h(α)− h(β)

α− β

)n f(β)

α− βdβ. (B.17)

Assume that f ∈ E−1w,T and g ∈ WN2−1,b. There is a constant C ′ ≥ 1 such that

〈t〉1/2−p0‖R0f‖HN0+1 + 〈t〉‖R0f‖WN2+1,2/5 + 〈t〉1/2−4p0‖SR0f‖HN1+1 ≤ C ′ε1‖f‖E−1w,T,

〈t〉‖R0g‖WN2+1,b ≤ C ′ε1〈t〉1/2‖g‖WN2−1,b , b ∈ [1/100, 2/5],(B.18)

for any t ∈ [0, T ]. Moreover ∥∥R1f∥∥HN0+1 ≤ (C ′ε1)2〈t〉p0−1‖f‖E−1

w,T,∥∥R1f

∥∥WN2+1,−1/10 ≤ (C ′ε1)2〈t〉−11/10‖f‖E−1

w,T,∥∥SR1f

∥∥HN1+1 ≤ (C ′ε1)2〈t〉4p0−1‖f‖E−1

w,T.

(B.19)

and, for any n ≥ 2,∥∥Rnf∥∥HN0+1 +∥∥SRnf∥∥HN1+1 ≤ (C ′ε1)n+1〈t〉−5/4‖f‖E−1

w,T. (B.20)

This is proved in subsection B.2 below. In rest of this subsection we show how to use it toprove Proposition B.1. We consider first a suitable decomposition of the operators Hγ .

Lemma B.4. Let

(H0f)(α) :=1

πip.v.

∫R

f(β)

α− βdβ, (B.21)

denote the unperturbed Hilbert transform, and consider the operators T1 and T2 defined by

(T1f)(α) :=1

πp.v.

∫R

h(α)− h(β)− h′(β)(α− β)

|γ(α)− γ(β)|2f(β) dβ,

(T2f)(α) :=1

πp.v.

∫R

h(α)− h(β)− h′(β)(α− β)

|γ(α)− γ(β)|2h(α)− h(β)

α− βf(β) dβ.

(B.22)

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WATER WAVES WITH SURFACE TENSION 81

ThenHγ = H0 − T1 + iT2, (B.23)

andT1 =

∑n≥0

(−1)nR2n, T2 =∑n≥0

(−1)nR2n+1. (B.24)

Moreover, if f ∈ E−1w,T and t ∈ [0, T ] then∥∥(T1 −R0)f

∥∥HN0+1 +

∥∥T2f∥∥HN0+1 . ε

21〈t〉p0−1‖f‖E−1

w,T,∥∥(T1 −R0)f

∥∥WN2+1,−1/10 +

∥∥T2f∥∥WN2+1,−1/10 . ε

21〈t〉−11/10‖f‖E−1

w,T,∥∥S(T1 −R0)f

∥∥HN1+1 +

∥∥ST2f∥∥HN1+1 . ε

21〈t〉4p0−1‖f‖E−1

w,T.

(B.25)

In particular‖Hγf‖E−1

w. ‖f‖E−1

w,T, (B.26)

‖R0f‖E1w + ‖T1f‖E1w + ‖T2f‖E1w . ε1〈t〉−1/2‖f‖E−1w,T. (B.27)

Proof. The identities (B.23) and (B.24) follow directly from definitions. The bounds (B.25)follow from (B.19)–(B.20). Notice that

H0g(ξ) = − sgn(ξ)g(ξ). (B.28)

The bounds (B.26)–(B.27) follow using (B.23) and the bounds (B.25) and (B.18).

We are now ready to define the conjugate pair (φ, ψ).

Lemma B.5. (i) We have

H2γ = I on E−1

w,T . (B.29)

Moreover, if φ ∈ E−1w,T is a real-valued function then there is a unique real-valued function

ψ ∈ E−1w,T with the property that

(I −Hγ)(φ+ iψ) = 0. (B.30)

(ii) The function F : Ω→ C,

F (z) :=1

2πi

∫R

(φ+ iψ)(β)γ′(β)

z − γ(β)dβ (B.31)

in a bounded analytic function in Ω, which extends to a C1 function in Ω with the propertythat F (x+ ih(x)) = (φ+ iψ)(x) for any x ∈ R.

Proof. The identity (B.29) is a standard consequence of the Cauchy integral formula appliedto the analytic function Ff defined in (B.13), and the second limit in (B.16).

The uniqueness of ψ satisfying (B.30) follows from Lemma B.4: if ψ1, ψ2 are real-valuedsolutions of (B.30) then

(I −Hγ)(ψ1 − ψ2) = 0.

Using the formula (B.30) and taking the real part, it follows that (I + T1)(ψ1−ψ2) = 0. Since‖T1‖E−1

w →E−1w. ε1 (see (B.27)), this shows that ψ1 − ψ2 = 0.

To prove existence we use that T1 is a contraction on E−1w,T and define ψ such that

(I + T1)ψ = (−iH0 + T2)φ. (B.32)

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82 ALEXANDRU D. IONESCU AND FABIO PUSATERI

Let P := (I − Hγ)(−iφ + ψ) and notice that (I + Hγ)P = 0 (as a consequence of (B.29)).Moreover, <P = 0, as a consequence of (B.32). Therefore P = 0. This completes the proof ofpart (i). The claims in (ii) follow from the second identity in (B.16).

Assume φ, ψ and F = G+ iH are as in Lemma B.5. Let

v1 := ∂xG = ∂yH, v2 := ∂yG = −∂xH.

Notice that

G(x+ ih(x)) = φ(x), H(x+ ih(x)) = ψ(x).

Therefore

−ψx = ∂yG(x+ ih(x))− ih′(x)∂xG(x+ ih(x)) = G(h)φ.

Let

V (x) := v1(x+ ih(x)), B(x) := v2(x+ ih(x)), x ∈ R.

Taking derivatives, we have φx = V + hxB and −ψx = B − hxV . Therefore

V =φx + hxψx

1 + h2x

= V, B =hxφx − ψx

1 + h2x

= B, (B.33)

where V,B are defined in (B.4). Notice that

∂xHγf = γxHγ(fx/γx).

Therefore the identity (B.30) gives

(I −Hγ)(φx + iψx

γx

)= 0.

Notice also that φx + iψx = (1 + ihx)(V − iB), as a consequence of (B.33). Therefore

(I −Hγ)(V − iB

)= 0.

To summarize, we have the following lemma:

Lemma B.6. Assume φ ∈ E1/2w,T and define ψ as in Lemma B.5. Then ψ ∈ E1/2

w,T and

‖ψ‖E1/2w,T

. ‖φ‖E1/2w,T

. (B.34)

Moreover,

G(h)φ = −ψx, V =φx + hxψx

1 + h2x

, B =hxφx − ψx

1 + h2x

. (B.35)

and

φx = V + hxB, −ψx = B − hxV, (I −Hγ)(V − iB

)= 0. (B.36)

In addition, for any t ∈ [0, T ] and any g ∈ φx, ψx,

〈t〉−p0‖g‖HN0−1/2,−1/10 + 〈t〉1/2‖g‖WN2−1/2,−1/10 + 〈t〉−4p0‖Sg‖HN1−1/2,−1/10 . ‖φ‖E1/2w,T

, (B.37)

and, for any f ∈ φx, ψx, V, B,

〈t〉−p0‖f‖HN0−1/2 + 〈t〉1/2‖f‖WN2−1/2 + 〈t〉−4p0‖Sf‖HN1−1/2 . ‖φ‖E1/2w,T

. (B.38)

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WATER WAVES WITH SURFACE TENSION 83

Proof. The inequality (B.34) follows from (B.27) and (B.32). The formulas (B.36) were derivedearlier. The bounds (B.37) follow as a consequence of (B.34) and the definition. The functionhx satisfies similar bounds, for any t ∈ [0, T ],

〈t〉−p0‖hx‖HN0,−1/2+p1 + 〈t〉1/2‖hx‖WN2,−1/10 + 〈t〉−4p0‖Shx‖HN1,−1/2+p1 . ε1, (B.39)

see (B.1). The desired bounds (B.38) for V,B follow from (B.35) and Lemma B.2.

We can now complete the proof of Proposition B.1.

Proof of Proposition B.1. We may assume ‖φ‖E1/2w,T

= 1. The bound (B.5) was already proved,

see (B.38). So we define G≥3 according to (B.6), and we need to prove the bounds (B.8).The last identity in (B.36) gives

(I −H0)(V − iB) + (T1 − iT2)(V − iB) = 0

Taking real and imaginary parts we have

V + iH0B = −T1V + T2B, −B + iH0V = T2V + T1B. (B.40)

Using also the formulas −ψx = B − hxV , see (B.36), and |∂x|φ = iH0(φx), we have

|∂x|φ−G(h)φ = iH0(φx)−B + hxV

= T2V + T1B + hxV + iH0(hxB).(B.41)

LetD := T2V + T1B = iH0V −B. (B.42)

Therefore

−G≥3 = |∂x|φ−G(h)φ− |∂x|TBh− ∂xTV h+G2 = I + II,

I : = hxV −H0(hxH0V )− |∂x|TiH0V h− ∂xTV h+G2 +R0(iH0V ),

II : = D −R0(iH0V )− iH0(hxD) + |∂x|TDh.(B.43)

Using the definitions and (B.56), we write

I(ξ) =1

∫Rh(η)V (ξ − η)p(ξ, η) dη + G2(ξ)

where

p(ξ, η) : = iη − iη sgn(ξ) sgn(ξ − η) + i|ξ| sgn(ξ − η)χ(ξ − η, η)− iξχ(ξ − η, η)

+ i(ξ − η)[1− sgn(ξ) sgn(ξ − η)]

= iξ[1− χ(ξ − η, η)][1− sgn(ξ) sgn(ξ − η)].

Using also the formulas (B.7) and (B.36), we have

I(ξ) =1

∫Rh(η)F(V − φx)(ξ − η)p(ξ, η) dη =

−1

∫Rh(η)F(hxB)(ξ − η)p(ξ, η) dη.

In view of (B.38)–(B.39), and Lemma B.2,

〈t〉1/2−p0‖hxB‖HN0−1/2 + 〈t〉‖hxB‖WN2−1/2 + 〈t〉1/2−4p0‖S(hxB)‖HN1−1/2 . ε1.

In other words, ε1hxB ∈ O2,−1/2, see Definition C.1, and hx ∈ O1,0, see (B.39). In view ofLemma C.2 (applied to m2(ξ, η) = p(ξ, η)/η), ε1I ∈ O3,1, which gives, for any t ∈ [0, T ],

〈t〉1−p0‖I‖HN0+1 + 〈t〉11/10‖I‖WN2+1 + 〈t〉1−4p0‖SI‖HN1+1 . ε2

1. (B.44)

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84 ALEXANDRU D. IONESCU AND FABIO PUSATERI

We consider now the contribution of II. It follows from (B.25) and (B.38) that∥∥(T1 −R0)B∥∥HN0+1 +

∥∥T2V∥∥HN0+1 . ε

21〈t〉p0−1,∥∥(T1 −R0)B

∥∥WN2+1,−1/10 +

∥∥T2V∥∥WN2+1,−1/10 . ε

21〈t〉−11/10,∥∥S(T1 −R0)B

∥∥HN1+1 +

∥∥ST2V∥∥HN1+1 . ε

21〈t〉4p0−1.

Therefore, using the formula (B.42),

〈t〉1−p0‖D−R0B‖HN0+1 +〈t〉11/10‖D−R0B∥∥WN2+1 +〈t〉1−4p0‖S(D−R0B)‖HN1+1 . ε2

1. (B.45)

Therefore, using also (B.18),

〈t〉1/2−p0‖D‖HN0+1 + 〈t〉‖D‖WN2+1,1/100 + 〈t〉1/2−4p0‖SD‖HN1+1 . ε1. (B.46)

Using (B.42), in the form R0D = R0(iH0V −B), and the bounds (B.18) and (B.46),

〈t〉1−p0‖R0B −R0(iH0V )‖HN0+1 + 〈t〉3/2‖R0B −R0(iH0V )‖WN2+1,1/100

+ 〈t〉1−4p0‖S[R0B −R0(iH0V )]‖HN1+1 . ε21.

(B.47)

Moreover,

F[− iH0(hxD) + |∂x|TDh

](ξ) =

1

∫RD(ξ − η)h(η) sgn(ξ)[ξχ(ξ − η, η)− η] dη.

Using (B.1) and (B.46), it follows from Lemma C.2∥∥− iH0(hxD) + |∂x|TDh∥∥HN0+1 . ε

21〈t〉−1+p0 ,∥∥− iH0(hxD) + |∂x|TDh

∥∥WN2+1 . ε

21〈t〉−11/10,∥∥S[−iH0(hxD) + |∂x|TDh]

∥∥HN1+1 . ε

21〈t〉−1+4p0 .

(B.48)

The desired bound (B.8) follows from (B.43) and the bounds (B.44)–(B.48).

B.2. Proof of Lemma B.3. We rewrite first the operators Rn. We take the Fourier transformin α and make the change of variables α→ β + ρ to write

F[Rnf

](ξ) =

1

π

∫R2

f(β)

ρ

(h(β + ρ)− h(β)

ρ

)nh(β + ρ)− h(β)− ρh′(β)

ρe−iξβe−iξρ dβdρ.

Notice thath(β + ρ)− h(β)

ρ=

1

∫Rh(η)eiηβ

eiηρ − 1

ρdη.

Therefore

F[Rnf

](ξ) =

1

π(2π)n+1

∫R2×Rn

f(β)

ρe−iξβe−iξρei(η1+...+ηn+1)β

× h(η1) · . . . · h(ηn+1)eiηn+1ρ − 1− iηn+1ρ

ρ

n∏l=1

eiηlρ − 1

ρdβdρdη1 . . . dηn.

This can be rewritten in the form

F[Rnf

](ξ) =

1

π(2π)n+1

∫Rn+1

Mn+1(ξ; η1, . . . , ηn+1)f(ξ − η1 − . . .− ηn+1)

× h(η1) · . . . · h(ηn+1) dη1 . . . dηn+1,

(B.49)

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WATER WAVES WITH SURFACE TENSION 85

where

Mn+1(ξ; η1, . . . , ηn+1) :=

∫R

e−iξρ

ρ

eiηn+1ρ − 1− iηn+1ρ

ρ

n∏l=1

eiηlρ − 1

ρdρ. (B.50)

Using the formula

d

e−iηρ − 1

ρ=

1− e−iηρ − iηρe−iηρ

ρ2,

and integration by parts in ρ in (B.50), we have

Mn+1(ξ; η1, . . . , ηn+1) = −∫R

e−iηn+1ρ − 1

ρ

d

[e−i(ξ−ηn+1)ρ

n∏l=1

eiηlρ − 1

ρ

]dρ

= −i(ξ − ηn+1)

∫Re−iξρ

n+1∏l=1

eiηlρ − 1

ρdρ

+n∑j=1

∫Re−iξρ

iηjρeiηjρ − eiηjρ + 1

ρ2

n+1∏l=1, l 6=j

eiηlρ − 1

ρdρ.

(B.51)

For j ∈ 1, . . . , n let πj : Rn+1 → Rn+1 denote the map that permutes the variables ηj andηn+1,

πj(η1, . . . , ηj , . . . , ηn, ηn+1) := (η1, . . . , ηn+1, . . . , ηn, ηj).

The formulas (B.50) and (B.51) show that

Mn+1(ξ; η) +n∑j=1

Mn+1(ξ;πj(η)) = −i(ξ − ηn+1)

∫Re−iξρ

n+1∏l=1

eiηlρ − 1

ρdρ

+

n∑j=1

∫Re−iξρ

iηjρeiηjρ − eiηjρ + 1

ρ2

n+1∏l=1, l 6=j

eiηlρ − 1

ρdρ

+n∑j=1

∫R

e−iξρ

ρ

eiηjρ − 1− iηjρρ

n+1∏l=1, l 6=j

eiηlρ − 1

ρdρ

= −i(ξ − η1 − . . .− ηn+1)

∫Re−iξρ

n+1∏l=1

eiηlρ − 1

ρdρ,

where η := (η1, . . . , ηn+1). Letting

Mn+1(ξ; η) : =1

n+ 1

[Mn+1(ξ; η) +

n∑j=1

Mn+1(ξ;πj(η))]

=−i(ξ − η1 − . . .− ηn+1)

n+ 1

∫Re−iξρ

n+1∏l=1

eiηlρ − 1

ρdρ,

(B.52)

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86 ALEXANDRU D. IONESCU AND FABIO PUSATERI

it follows by symmetrization from (B.49) that

F[Rnf

](ξ) =

1

π(2π)n+1

∫Rn+1

Mn+1(ξ; η)f(ξ − η1 − . . .− ηn+1)

× h(η1) · . . . · h(ηn+1) dη.

(B.53)

B.2.1. The operator R0. We prove now the bounds (B.18). Recall the formulas∫Re−iξx dx = 2πδ0(ξ),

∫Re−iξx

1

xdx = −iπ sgn(ξ),

∫R

e−iξx − 1

x2dx = −π |ξ|, (B.54)

for any ξ ∈ R. Using these formulas, the symbol M1(ξ; η1) can be calculated easily,

M1(ξ; η1) = π(ξ − η1)[

sgn(ξ)− sgn(ξ − η1)]. (B.55)

Therefore

F[R0f

](ξ) =

1

∫R

(ξ − η1)[

sgn(ξ)− sgn(ξ − η1)]f(ξ − η1)h(η1) dη1. (B.56)

The bounds (B.18) can be proved easily. We may assume that ‖f‖E−1w,T. 1. It follows from

(B.56) that

‖PkR0f‖L2 .∑

k2+10≥max(k,k1)

2k1‖P ′k1f‖L∞‖P′k2h‖L2 . 〈t〉−1/2

∑k2+10≥k

23 min(k2,0)/5‖P ′k2h‖L2 ,

for any k ∈ Z. Moreover,

‖R0f‖L2 .∑

k2+10≥k1

2k1‖P ′k1f‖L∞‖P′k2h‖L2 . ε1〈t〉p0−1/2.

The L2 bound on R0f in (B.18) follows using the assumptions (B.1). The L∞ bound on R0gin the second line of (B.18) follows in a similar way, and this implies the L∞ bound on R0f(t)in the first line of (B.18) (by setting b = 2/5). The weighted bound in (B.18) also follows inthe same way, using (2.5) and the homogeneity of the symbol defining R0.

B.2.2. The operator R1. We prove now the bounds (B.19). Using (B.54) again

M2(ξ; η1, η2) :=iπ

2(ξ − η1 − η2)

[|ξ − η1 − η2|+ |ξ| − |ξ − η1| − |ξ − η2|

]. (B.57)

The main observation is that M2(ξ; η1, η2) = 0 if |η1|+ |η2| ≤ |ξ|. It is easy to see that∥∥∥F−1[M2(ξ; η1, η2)ϕk1(η1)ϕk2(η2)ϕk3(ξ − η1 − η2)

]∥∥∥L1(R3)

. 2k32min(k1,k2).

Therefore, using Lemma 2.1, the assumption ‖f‖E−1w,T. 1, and the bounds (B.1)

‖PkR1f‖Lp .∑

k2+10≥max(k,k1,k3)

2k1+k3‖P ′k1h‖L∞‖P′k2h‖Lp‖P

′k3f‖L∞

. ε1〈t〉−1∑

k2+10≥k23 min(k2,0)/5‖P ′k2h‖Lp ,

(B.58)

for p ∈ 2,∞ and any k ∈ Z. In particular,

‖P≥0R1f‖HN0+1 . ε21〈t〉−1+p0 and ‖P≥0R1f‖WN2+1,−1/10 . ε2

1〈t〉−3/2. (B.59)

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WATER WAVES WITH SURFACE TENSION 87

Moreover

‖R1f‖L2 .∑

k2+10≥max(k1,k3)

2k1+k3‖P ′k1h‖L∞‖P′k2h‖L2‖P ′k3f‖L∞ . ε

21〈t〉p0−1. (B.60)

Finally, as a consequence of (B.1), (B.58), and Sobolev embedding,

2−k/10‖PkR1f‖L∞ . 22k/5‖PkR1f‖L2 . 22k/5ε21〈t〉p0−1,

2−k/10‖PkR1f‖L∞ . 2−k/10ε1〈t〉−1 · 2−2k/5ε1〈t〉−1/2 . ε21〈t〉−3/22−k/2,

(B.61)

for any integer k ≤ 0. The bounds in the first two lines of (B.19) follow from (B.59)–(B.61).The weighted bound in the last line follows in a similar way, using an identity similar to (2.5)and the homogeneity of the symbol.

B.2.3. The operator Rn, n ≥ 2. To prove (B.20) we would like to use induction over n. Forthis we need to prove slightly stronger bounds. We start with a lemma:

Lemma B.7. For any k = (k1, . . . , kn+1) ∈ Zn+1 and n ≥ 1 let Fn;k(fk) be defined by

F(Fn;k(fk)

)(ξ) :=

∫Rn+1

M ′n+1(ξ; η)ϕk1(η1) . . . ϕkn+1(ηn+1)

× fk(ξ − η1 − . . .− ηn+1) · h(η1) · . . . · h(ηn+1) dη,

(B.62)

where η = (η1, . . . , ηn+1) and

M ′n+1(ξ; η) :=

∫Re−iξρ

n+1∏l=1

eiηlρ − 1

ρdρ. (B.63)

Assume that the functions fk satisfy the uniform bounds

‖fk‖W 1/5,0 ≤ 〈t〉−1/2. (B.64)

Then there is a constant C0 ≥ 1 such that if n ≥ 2 then∥∥∥ ∑k∈Zn+1

Fn;k(fk)∥∥∥HN0+1

≤ (C0ε1)n+1〈t〉−5/4. (B.65)

Proof. Notice that

M ′n+1(ξ, η1, . . . , ηn+1) = 0 if η1 · . . . · ηn+1 = 0. (B.66)

Taking partial derivatives in η it follows that

M ′n+1(ξ, η1, . . . , ηn+1) = 0 if |η1|+ . . .+ |ηn+1| ≤ |ξ|. (B.67)

Let χ0 := F−1(ϕ0) and let χ′0 denote the derivative of χ0. Let k1 ≤ . . . ≤ kn+1 denote theincreasing rearrangement of the integers k1, . . . , kn+1. In view of (B.67),

Fn,k(fk) = Fn,k(P≤kn+1+5nfk). (B.68)

Using the formula

1

∫Rϕk(µ)eiyµ

eiµρ − 1

ρdµ =

2k[χ0(2k(y + ρ))− χ0(2ky)

=: Kk(y, ρ), (B.69)

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88 ALEXANDRU D. IONESCU AND FABIO PUSATERI

and the definition (B.62), we rewrite

Fn,k(fk)(x) = (2π)n+1

∫Rn+2

fk(x− ρ)n+1∏l=1

P ′klh(x− ρ− yl) ·n+1∏l=1

Kkl(yl, ρ) dρdy. (B.70)

We notice that ∫R

∣∣Kk(y, ρ)∣∣ dy . min

(|ρ|−1, 2k

). (B.71)

The inequalities (B.71) show that∫Rn+2

∣∣∣ n+1∏l=1

Kkl(yl, ρ)∣∣∣ dρdy ≤ Cn+12k1+...+kn(1 + |kn+1 − kn|), (B.72)

for some constant C ≥ 1 (which is allowed to change from now on from line to line). Therefore,using (B.70),∥∥Fn,k(fk)∥∥L2 ≤ Cn+1(1 + |kn+1 − kn|) · ‖fk‖L∞‖P ′kn+1

h‖L2

n∏l=1

2kl‖P ′klh‖L∞ . (B.73)

Therefore ∑(k1,...,kn+1)∈J1

∥∥Fn,k(fk)∥∥HN0+1 ≤ (Cε1)n+1〈t〉−(n+1)/2+p0 ,

∑(k1,...,kn+1)∈J2

∥∥Fn,k(P≥kn+1−10n+1fk)∥∥HN0+1 ≤ (Cε1)n+1〈t〉−(n+1)/2+p0 ,

(B.74)

where

J1 :=

(k1, . . . , kn+1) ∈ Zn+1 : kn+1 ≤ max(0, kn) + 10n,

J2 :=

(k1, . . . , kn+1) ∈ Zn+1 : k1 = kn+1 ≥ max(0, kn) + 10n.

(B.75)

It remains to prove that if n ≥ 1 then∥∥∥ ∑(k1,...,kn+1)∈J2

Fn,k(P≤k1−10nfk)∥∥∥HN0+1

≤ (C1ε1)n+1〈t〉−en (B.76)

for some constant C1 ≥ 1, where e1 = 1− p0 and en = 5/4 if n ≥ 2.We prove the inequalities in (B.76) using induction over n (the case n = 1 follows from

our discussion of the operator R1 before, noticing that only L∞ norms on ∂xf were used in(B.58)–(B.60)). We decompose

Kkn+1(yn+1, ρ) = Kkn+1(yn+1, ρ) + 22kn+1χ′0(2kn+1yn+1)

where

Kk(y, ρ) :=2k[χ0(2k(y + ρ))− χ0(2ky)− 2kρχ′0(2ky)

.

Then we decompose

Fn,k(P≤k1−10nfk) = F 1n,k(P≤k1−10nfk) + F 2

n,k(P≤k1−10nfk),

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WATER WAVES WITH SURFACE TENSION 89

where

F 1n,k(g)(x) := Cn+1

∫Rn+2

g(x−ρ)n+1∏l=1

P ′klh(x−ρ−yl)·Kkn+1(yn+1, ρ)n∏l=1

Kkl(yl, ρ) dρdy, (B.77)

and

F 2n,k(g)(x) := Cn+1

∫Rn+2

g(x− ρ)

n+1∏l=1

P ′klh(x− ρ− yl)

× 22kn+1χ′0(2kn+1yn+1)n∏l=1

Kkl(yl, ρ) dρdy.

(B.78)

We notice that ∫R

∣∣Kk(y, ρ)∣∣ dy . 2kρmin

(|ρ|−1, 2k

).

Using also (B.71) it follows that if (k1, . . . , kn+1) ∈ J2 then∫Rn+2

∣∣∣Kkn+1(yn+1, ρ)n∏l=1

Kkl(yl, ρ)∣∣∣ dρdy ≤ Cn+1(1 + |kn − kn−1|)2k1+...+kn ,

which is slightly stronger than the inequality (B.72). Therefore∥∥∥F 1n,k(P≤k1−10nfk)

∥∥∥L2≤ Cn+1(1 + |kn − kn−1|) · ‖fk‖L∞‖P ′k1h‖L2

n∏l=1

2kl‖P ′klh‖L∞ .

Therefore, for any l ≥ 0,∥∥∥Pl[ ∑k∈J2

F 1n,k(P≤k1−10nfk)

]∥∥∥L2≤ Cn+1(ε1〈t〉−1/2)n〈t〉−1/2

∑|k1−l|≤10

‖P ′k1h‖L2 .

Therefore ∥∥∥ ∑k∈J2

F 1n,k(P≤k1−10nfk)

∥∥∥HN0+1

≤ (Cε1)n+1〈t〉−5/4. (B.79)

We estimate now the contributions of the terms F 2n,k(P≤k1−10nfk). We integrate the variable

yn+1 in the defining formula (B.78). Let

g(k1,...,kn)(z) :=∑

kn+1≤k1−10n

f(k1,...,kn,kn+1)(z)

∫RP ′kn+1

h(z − yn+1)22kn+1χ′0(2kn+1yn+1) dyn+1.

Using the assumptions (B.64) and (B.1), it is easy to see that∑l∈Z

(2l/5 + 1)‖Plg(k1,...,kn)‖L∞ . 〈t〉−1/2 · ε1〈t〉−2/5.

The induction hypothesis shows that∥∥∥ ∑k∈J2

F 2n,k(P≤kn+1−10n

fk)∥∥∥HN0+1

≤ (C1ε1)n〈t〉−1+p0 · Cε1〈t〉−2/5.

The desired conclusion follows, using also (B.79) provided that C1 is sufficiently large.

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90 ALEXANDRU D. IONESCU AND FABIO PUSATERI

The inequality on the first term in (B.20) follows from this lemma with fk = ∂xf . To provethe bound on the second term we start from the formula (B.53) and write, as in (2.5),

F[SRnf

](ξ) =

cn(2π)n+1

∫Rn+1

M ′n+1(ξ; η)S∂xf(ξ − η1 − . . .− ηn+1)h(η1) . . . h(ηn+1) dη

+(n+ 1)cn(2π)n+1

∫Rn+1

M ′n+1(ξ; η)∂xf(ξ − η1 − . . .− ηn+1)h(η1) . . . Sh(ηn+1) dη

+cn

(2π)n+1

∫Rn+1

M ′n+1(ξ; η)∂xf(ξ − η1 − . . .− ηn+1)h(η1) . . . h(ηn+1) dη,

(B.80)

where cn := −1/(π(n+ 1)), M ′n+1(ξ; η) is as in (B.63), and

M ′n+1(ξ; η) := −(ξ∂ξ +

n+1∑j=1

ηj∂ηj

)(M ′n+1

)(ξ; η).

We notice that M ′n+1(λξ;λη) = λnM ′n+1(ξ; η), if λ > 0. Taking the λ derivative it follows that

M ′n+1(ξ; η) = −nM ′n+1(ξ; η). (B.81)

The estimate on SRnf in (B.20) follows by the same argument as in the proof of LemmaB.7, using dyadic decompositions and the bounds (B.1). As a general rule, we always estimatethe factor that has the vector-field S in L2 and the remaining factors in L∞. This completesthe proof of Lemma B.3.

Appendix C. Elliptic bounds

In this appendix we prove elliptic-type bounds on several multilinear expressions that appearin the course of the proofs, mainly in the derivation of the equations done in section 3, and inthe energy estimates in sections 4-7.

C.1. The spaces Om,α. We start with a definition (recall (2.2)).

Definition C.1. Assume α ∈ [−2, 2] and let b := −1/10. Let O1,α denote the set of functionsf1 ∈ C([0, T ] : L2) that satisfy the “linear” bounds, for any t ∈ [0, T ],

〈t〉−p0∥∥f1(t)

∥∥HN0+α,b

+ 〈t〉−4p0∥∥Sf1(t)

∥∥HN1+α,b

+ 〈t〉1/2‖f1(t)‖WN2+α,b . ε1. (C.1)

Let O2,α denote the set of functions f2 ∈ C([0, T ] : L2) that satisfy the “quadratic” bounds

〈t〉1/2−p0∥∥f2(t)

∥∥HN0+α

+ 〈t〉1/2−4p0∥∥Sf2(t)

∥∥HN1+α

+ 〈t〉‖f2(t)‖WN2+α

. ε21. (C.2)

Let O3,α denote the set of functions f3 ∈ C([0, T ] : L2) that satisfy the “cubic” bounds

〈t〉1−p0∥∥f3(t)

∥∥HN0+α

+ 〈t〉1−4p0∥∥Sf3(t)

∥∥HN1+α

+ 〈t〉11/10‖f3(t)‖WN2+α

. ε31. (C.3)

In other words, the generic notation Om,α measures (1) the degree of the function (linear,quadratic, or cubic) represented by the exponent m, and (2) the number of derivatives under

control, relative to the Hamiltonian variables |∂x|h, |∂x|1/2φ which correspond to α = 0. Noticethat O3,α ⊆ O2,α; in the linear case O1,α we make slightly stronger assumptions on the lowfrequency part of the L2 norms.

We will often use the following lemma to estimate products and paraproducts of functions.

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WATER WAVES WITH SURFACE TENSION 91

Lemma C.2. Assume that m1,m2 : R× R→ C are continuous functions with

suppm1 ⊆ (ξ, η) ∈ R2 : |ξ − η| ≤ |η|/24,suppm2 ⊆ (ξ, η) ∈ R2 : |ξ − η|/|η| ∈ [2−20, 220].

(C.4)

Assume that∥∥F−1[m1(ξ, η)ϕk(ξ)]∥∥L1 . 1,

∥∥F−1[m2(ξ, η)ϕk1(ξ − η)ϕk2(η)]∥∥L1 . 1, (C.5)

for any k, k1, k2 ∈ Z. Let ml := −(ξ∂ξ + η∂η)ml, see (2.6), and assume that the multipliersm1, m2 satisfy the bounds (C.5) as well. Let M1 and M2 denote the bilinear operators associatedto m1 and m2, see (2.4). Assume that m,n ∈ 1, 2, 3. Then, for any α ∈ [−2, 2],

if f ∈ Om,−2, g ∈ On,α then M1(f, g) ∈ Omin(m+n,3),α ∩O1,α, (C.6)

and, for any α ∈ [−2, 0],

if f ∈ Om,α, g ∈ On,α then M2(f, g) ∈ Omin(m+n,3),α+2 ∩O1,α+2. (C.7)

Proof. We only show in detail how to prove the bounds (C.6) and (C.7) when m = n = 1, sincethe other cases are similar. We assume that t ∈ [0, T ] is fixed and sometimes drop it from thenotation. Using Lemma 2.1, the definition (C.1), and the assumption (C.5),∥∥PkM1(f, g)

∥∥Lp. ‖P ′kg‖Lp‖P≤kf‖L∞ . ε1〈t〉−1/2‖P ′kg‖Lp

for p ∈ 2,∞ and for any k ∈ Z. Therefore∥∥M1(f, g)∥∥HN0+α,b

. ε21〈t〉−1/2+p0 ,∥∥M1(f, g)

∥∥WN2+α,b

. ε21〈t〉−1.

(C.8)

Similarly, using Lemma 2.1, the definition (C.1), and the assumption (C.5),∥∥PkM1(f, Sg)∥∥L2 . ‖P ′kSg‖L2‖P≤kf‖L∞ . ε1〈t〉−1/2‖P ′kSg‖L2 ,

and ∥∥PkM1(Sf, g)∥∥L2 . ‖P ′kg‖L∞‖P≤kSf‖L2 . ε1〈t〉4p0ε1〈t〉−1/2 min(2−2bk, 2−k(N2+α)).

Therefore ∥∥M1(f, Sg)∥∥HN1+α,b

+∥∥M1(Sf, g)

∥∥HN1+α,b

. ε21〈t〉−1/2+4p0 . (C.9)

Moreover, since m1 satisfies the same bounds as m1, we have as in the proof of (C.8),∥∥M1(f, g)∥∥HN1+α,b

. ε21〈t〉−1/2+4p0 . (C.10)

The desired identities (C.6) follow from the bounds (C.8)–(C.10) and the identity (2.5).We consider now the operator M2 and prove (C.7) when m = n = 1. We estimate first,

using as before Lemma 2.1, the definition (C.1), and the assumption (C.5),∥∥M2(f, g)∥∥L2 .

∑|k1−k2|≤30

‖Pk1f‖L2‖Pk2g‖L∞ . ε21〈t〉−1/2+p0 (C.11)

and, similarly, ∥∥M2(f, g)∥∥L∞. ε2

1〈t〉−1, (C.12)∥∥M2(f, g)∥∥L1 . ε

21〈t〉2p0 , (C.13)∥∥M2(Sf, g)

∥∥L2 +

∥∥M2(f, Sg)∥∥L2 +

∥∥M2(f, g)∥∥L2 . ε

21〈t〉−1/2+4p0 , (C.14)

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92 ALEXANDRU D. IONESCU AND FABIO PUSATERI∥∥M2(Sf, g)∥∥L1 +

∥∥M2(f, Sg)∥∥L1 +

∥∥M2(f, g)∥∥L1 . ε

21〈t〉8p0 . (C.15)

These estimates and Sobolev embedding (in the form ‖Pkh‖L2 . 2k/2‖Pkh‖L1) provide thedesired estimates on low frequencies,

〈t〉1/2−p0∥∥P≤4M2(f, g)

∥∥HN0+2+ 〈t〉1/2−4p0

∥∥P≤4SM2(f, g)∥∥HN1+2+ 〈t〉‖P≤4M2(f, g)‖

WN2+2 . ε21

and

〈t〉−p0∥∥P≤4M2(f, g)

∥∥HN0+2,b + 〈t〉−4p0

∥∥P≤4SM2(f, g)∥∥HN1+2,b

+ 〈t〉1/2‖P≤4M2(f, g)‖WN2+2,b . ε21.

To estimate the high frequencies we notice that, for k ≥ 0 and p ∈ 2,∞∥∥PkM2(f, g)∥∥Lp.

∑k1≥k−30

∥∥Pk1f∥∥Lp · ε1〈t〉−1/22−k1(N2+α).

Therefore ∥∥P≥0M2(f, g)∥∥HN0+α+2 . ε

21〈t〉−1/2+p0 ,∑

k≥0

2(N2+α+2)k∥∥PkM2(f, g)

∥∥L∞. ε2

1〈t〉−1. (C.16)

Similarly∥∥P≥0M2(Sf, g)∥∥HN1+α+2 +

∥∥P≥0M2(f, Sg)∥∥HN1+α+2 +

∥∥P≥0M2(f, g)∥∥HN1+α+2 . ε

21〈t〉−1/2+4p0 .

The desired conclusions in (C.7) follow.

C.2. Linear, quadratic, and cubic bounds. In this subsection we prove the elliptic boundson the quadratic and the cubic terms used implicitly to justify the calculations in section 3.Recall the formulas derived in section 3,

σ = (1 + h2x)−3/2 − 1, γ = (1 + h2

x)−3/4 − 1, p1 = γ, p0 = −(3/4)γx,

B =G(h)φ+ hxφx

(1 + h2x)

, V = φx −Bhx, ω = φ− TBP≥1h.(C.17)

Recall the bounds (2.25). With the notation in Definition C.1, and using also Proposition B.1and Lemma C.2, we have

hx ∈ O1,0, φx ∈ O1,−1/2, G(h)φ ∈ O1,−1/2, B ∈ O1,−1/2, V ∈ O1,−1/2. (C.18)

Using (C.18) and Lemma C.2, it follows that

σ ∈ O2,0, γ ∈ O2,0, p1 ∈ O2,0, p0 = O2,−1,

ωx − φx ∈ O2,0, |∂x|ω − |∂x|φ ∈ O2,0, V − φx ∈ O2,−1/2, B −G(h)φ ∈ O2,−1/2.(C.19)

Using Proposition B.1 it follows that

G(h)φ− |∂x|φ ∈ O2,0, B − |∂x|ω ∈ O2,−1/2, G2(h, φ) ∈ O2,1, G≥3 ∈ O3,1. (C.20)

The main evolution system (2.24) shows that

∂th ∈ O1,−1/2, ∂tφ ∈ O1,−1.

The formula (B.32), that is (I + T1)ψ = (−iH0 + T2)φ, now shows that

∂t(G(h)φ)− iH0∂tφx ∈ O2,−2.

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WATER WAVES WITH SURFACE TENSION 93

Using again the main system (2.24) and previous identities,

∂th− |∂x|φ ∈ O2,0, ∂tφ− hxx ∈ O2,−1, ∂t(G(h)φ) + |∂x|3h ∈ O2,−2,

∂tB + |∂x|3h ∈ O2,−2, ∂tV − ∂3xh ∈ O2,−2.

(C.21)

Finally, the formulas (3.17) and (3.21) show that

u ∈ O1,0, N2(h, ω) ∈ O2,0, V − i

2∂x|∂x|−1/2(u− u) ∈ O2,−1/2. (C.22)

The following proposition provides suitable bounds on the cubic remainders.

Proposition C.3. Let (G≥3,Ω≥3), U≥3, OWk, and OZk denote the cubic remainders in Propo-

sition 3.1, Proposition 3.4, Proposition 3.5, and Lemma 3.6 respectively. Then, with N = 3k/2,

G≥3 ∈ O3,1, Ω≥3 ∈ O3,1/2, U≥3 ∈ |∂x|1/2O3,1/2, (C.23)

〈t〉1−p0‖OW (t)‖HN0−N,−1/2 + 〈t〉1−4p0‖SOW (t)‖HN1−N,−1/2 . ε31, if N ≤ N1,

〈t〉1−p0‖OW (t)‖HN0−N,−1/2 . ε31, if N ∈ [N1, N0],

(C.24)

and

〈t〉1−4p0∥∥OZ(t)

∥∥H0,−1/2 . ε

31. (C.25)

Proof. The desired conclusion G≥3 = O3,1 was already proved in Proposition B.1. We examinenow the term Ω≥3 in Proposition 3.1. Inspecting the calculations that lead from (3.10) to(3.11), we see that

Ω≥3 =

5∑j=1

Ωj≥3,

Ω1≥3 :=

∂2xh

(1 + h2x)3/2

−(hxx + ∂xTσ∂xh

),

Ω2≥3 :=

[R(B,B)−R(|∂x|ω, |∂x|ω)

]/2−

[R(V, V )−R(∂xω, ∂xω)

]/2−R(V,Bhx),

Ω3≥3 := −T∂tBP≥1h− T|∂x|3hP≥1h,

Ω4≥3 := TBP≤0G(h)φ− T|∂x|ωP≤0|∂x|ω,

Ω5≥3 := TB(V hx)− TBhxV − TV ∂x(TBP≥1h).

(C.26)

We examine the terms and use (C.18)–(C.21) and Lemma C.2 to see

Ω1≥3 = O3,1, Ω2

≥3 = O3,1, Ω3≥3 = O3,1, Ω4

≥3 = O3,1.

Moreover, an argument similar to the proof of (C.6) in Lemma C.2 shows that

Tf1f2g − Tf1Tf2g = O3,1 if f1 = O1,−1, f2 = O1,−1, g = O1,−1. (C.27)

Therefore, we can rewrite

Ω5≥3 =

[TBThxV − TBhxV

]+ TB

(R(V, hx)

)+[TBTV hx − TV TBP≥1hx

]− TV T∂xBP≥1h,

and conclude that Ω5≥3 = O3,1. Therefore Ω≥3 = O3,1 as desired.

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94 ALEXANDRU D. IONESCU AND FABIO PUSATERI

We now move on to examine the cubic terms appearing in (3.42) in Proposition 3.4. For thiswe inspect the computations in the proof of Proposition 3.2 to retrieve all cubic terms thathave been incorporated in U≥3. We find

U≥3 :=5∑j=1

U j≥3, (C.28)

where

U1≥3 := |∂x|G≥3 − i|∂x|1/2Ω≥3 + T∂tp1P≥1|∂x|h+ T∂tp0P≥1|∂x|−1∂xh

+ Tp1P≥1|∂x|(∂th− |∂x|ω + ∂xTV h

)+ Tp0P≥1|∂x|−1∂x

(∂th− |∂x|ω + ∂xTV h

),

(C.29)

U2≥3 := −

[Tp1P≥1|∂x|, ∂xTV

]h−

[Tp0P≥1|∂x|−1∂x, ∂xTV

]h+

[|∂x|, ∂xT−V+∂xω

]h

+ iT−∂xV+∂2xω|∂x|1/2ω + i

[|∂x|1/2, ∂xTV−∂xω

]ω,

(C.30)

U3≥3 := i|∂x|3/2TσP≤0|∂x|h+ i

(|∂x|3/2TσP≥1|∂x|h− TσP≥1|∂x|5/2h+

3

2T∂xσP≥1|∂x|1/2∂xh

)− i(|∂x|3/2Tp1P≥1|∂x|h− Tp1P≥1|∂x|5/2h+

3

2T∂xp1P≥1|∂x|1/2∂xh

)− i(|∂x|3/2Tp0P≥1|∂x|−1∂xh− Tp0P≥1|∂x|1/2∂xh

),

(C.31)

U4≥3 := −3i

4T∂xγP≥1|∂x|−1/2∂xTp0 |∂x|−1∂xh

− i(TγP≥1|∂x|3/2Tp1P≥1|∂x|h− Tγp1P≥1|∂x|5/2h+

3

2Tγ∂xp1P≥1|∂x|1/2∂xh

)− i(TγP≥1|∂x|3/2Tp0P≥1|∂x|−1∂xh− Tγp0P≥1|∂x|1/2∂xh

)+

3i

4

(T∂xγP≥1|∂x|−1/2∂xTp1P≥1|∂x|h− T∂xγp1P≥1|∂x|1/2∂xh

),

(C.32)

U5≥3 := [|∂x|, ∂xT∂xω](h− h) + i|∂x|1/2[T|∂x|3hP≥1h− T|∂x|3hP≥1h] + |∂x|M2(ω, h− h), (C.33)

where h = (1/2)|∂x|−1(u+u), see (3.30). The first term comes from the remainders in the firstchain of identities in the proof of Proposition 3.2. This is the same remainder that appears alsoat the end of (3.22). The second term above comes from (3.23). The term U3

≥3 contains cubic

terms that are discarded in (3.25). Quartic terms that have been discarded in (3.25), and thelast term on the right-hand side of (3.24), are in U4

≥3. The term U5≥3 comes from the formulas

after (3.30).We examine the formulas (C.28)–(C.32) and use Lemma C.2 and (C.27) to conclude that

U≥3 = |∂x|1/2O3,1/2, as desired. This completes the proof of (C.23).We turn now to the proof of (C.24). We examine the formula (3.54) and observe that

[∂t,Dk

]=

k−1∑j=0

Dj [∂t,D]Dk−j−1 =

k−1∑j=0

Dj [∂t,Σγ ]Dk−j−1. (C.34)

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WATER WAVES WITH SURFACE TENSION 95

In view of (C.21) and Lemma C.2, [∂t,Σγ ] is an operator of order 3/2 that transforms linearterms into cubic terms. Moreover

Dk − |∂x|N =

′∑P1,...,Pk∈|∂x|3/2,Σγ

P1 . . . Pk,

where the sum above is taken over all possible choices of operators P1, . . . , Pk ∈ |∂x|3/2,Σγ,not all of them equal to |∂x|3/2. Therefore Dk − |∂x|N is an operator of order 3k/2 thattransforms linear terms into cubic terms. Finally, using Lemma A.1, Lemma A.2, and LemmaC.2, Nu is a quadratic term that does not lose derivatives, Nu ∈ O2,0. The desired conclusionfollows by applying elliptic estimates, as in the proof of Lemma C.2, to the terms in the lasttwo lines of (3.54).

The proof of (C.25) is similar, using the formula (3.63), Lemma C.2, and (C.34).

The next lemma gives bounds on the nonlinear terms in Propositions 3.4, 3.5 and 3.6.

Lemma C.4. With the notation in definition C.1 we have

∂tu− iΛu ∈ |∂x|1/2O2,−1, (C.35)

Nu ∈ |∂x|3/2O2,3/2. (C.36)

Moreover

‖Pk(∂t − iΛ)u‖L2 . ε212k/2(1 + t)−1/2+p0(2k/2 + (1 + t)−1/2),

‖Pk(∂t − iΛ)Su‖L2 . ε212k/2(1 + t)−1/2+4p0(2k/2 + (1 + t)−1/2).

(C.37)

Furthermore, with notation of Proposition 3.6, let Z = SD2N1/3u and

NZ := NZ,1 +NZ,2 +NZ,3. (C.38)

Then, we have

‖Pk(∂t − iΛ)Z‖L2 . ε212k/22max(k,0)(1 + t)−1/2+4p0 , (C.39)

‖PkNZ‖L2 . ε212min(k,0)(1 + t)−1/2+4p0 . (C.40)

Proof. The bounds (C.35)–(C.36) follow from the formulas (3.42)–(3.44), the bounds on thesymbols in Lemmas A.1 and A.2, and Lemma C.2. The bounds (C.37), which provide a morerefined version of the bounds (C.35) when k ≤ 0, follow by decomposing the nonlinearity intoquadratic and cubic components.

The estimates (C.39)–(C.40) follow from the formulas in Proposition 3.6, using the samesymbols bounds as before and Lemma 2.1.

C.3. Semilinear expansions. Recall the notation for trilinear operators

F[M(f1, f2, f3)

](ξ) :=

1

4π2

∫∫R×R

m(ξ, η, σ)f1(ξ − η)f2(η − σ)f3(σ) dηdσ. (C.41)

The starting point for the semilinear analysis in sections 8 and 9 leading to pointwise decay,is the equation (8.2) stated at the beginning of section 8. In this subsection we show how toderive (8.2) starting from the main equation (2.24) in Proposition 2.4. In particular, to derive(8.2) we will need to expand the nonlinearity as a sum of quadratic, cubic, and higher order

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96 ALEXANDRU D. IONESCU AND FABIO PUSATERI

terms. We point out that at this point we are not interested in the potential loss of derivatives.The following is the main result in this subsection.

Lemma C.5. Let U = |∂x|h− i|∂x|1/2φ, where (h, φ) are as in Proposition 2.4. Then we canwrite

∂tU − i|∂x|3/2U = QU + CU +R≥4, (C.42)

where the following claims hold true:

• The quadratic nonlinear terms are

QU =∑

(ε1ε2)∈(++),(+−),(−−)

Qε1ε2(Uε1 , Uε2), (C.43)

where U+ = U , U− = U , and the operators Q++, Q+−, Q−− are defined by the symbols

q++(ξ, η) :=i|ξ|(ξη − |ξ||η|)8|η|1/2|ξ − η|

+i|ξ|(ξ(ξ − η)− |ξ||ξ − η|)

8|η||ξ − η|1/2+i|ξ|1/2(η(ξ − η) + |η||ξ − η|)

8|η|1/2|ξ − η|1/2,

q+−(ξ, η) := − i|ξ|(ξη − |ξ||η|)4|η|1/2|ξ − η|

+i|ξ|(ξ(ξ − η)− |ξ||ξ − η|)

4|η||ξ − η|1/2− i|ξ|1/2(η(ξ − η) + |η||ξ − η|)

4|η|1/2|ξ − η|1/2,

q−−(ξ, η) := − i|ξ|(ξη − |ξ||η|)8|η|1/2|ξ − η|

− i|ξ|(ξ(ξ − η)− |ξ||ξ − η|)8|η||ξ − η|1/2

+i|ξ|1/2(η(ξ − η) + |η||ξ − η|)

8|η|1/2|ξ − η|1/2.

(C.44)

• The cubic terms have the form

CU := M++−(U,U, U) +M+++(U,U, U) +M−−+(U,U, U) +M−−−(U,U,U), (C.45)

with purely imaginary symbols mι1ι2ι3 such that∥∥F−1[mι1ι2ι3(ξ, η, σ) · ϕk(ξ)ϕk1(ξ − η)ϕk2(η − σ)ϕk3(σ)

]∥∥L1 . 2k/22max(k1,k2,k3) (C.46)

for all (ι1ι2ι3) ∈ (+ +−), (−−+), (+ + +), (−−−). Moreover, with d1 = 1/16,

m++−(ξ, 0,−ξ) = id1|ξ|3/2. (C.47)

• R≥4 is a quartic remainder satisfying

‖R≥4‖L2 + ‖SR≥4‖L2 . ε41〈t〉−5/4. (C.48)

Moreover

CU +R≥4 ∈ |∂x|1/2O3,−1. (C.49)

Let O4 denote the set of functions g ∈ C([0, T ] : L2) that satisfy the ”quartic” bounds

‖g(t)‖L2 + ‖Sg(t)‖L2 . ε41〈t〉−5/4, (C.50)

for any t ∈ [0, T ], compare with (C.48). It is easy to see, using the same argument as in theproof of Lemma C.2, that

M(O1,−2, O3,−2) ⊆ O4, M(O2,−2, O2,−2) ⊆ O4 (C.51)

for any bilinear operator M associated to a multiplier m satisfying

‖mk,k1,k2‖S∞ + ‖mk,k1,k2‖S∞ . 1, for any k, k1, k2 ∈ Z, (C.52)

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WATER WAVES WITH SURFACE TENSION 97

where m(ξ, η) = −(ξ∂ξ + η∂η)m(ξ, η).To identify the cubic terms we need the following more precise expansion of the Dirichlet–

Neumann map. This follows from the formula (B.32) (recall that G(h)φ = −ψx) and the proofof Lemma B.4 (see, in particular, (B.20), (B.53), (B.55), and (B.57)).

Lemma C.6. With (h, φ) as in Proposition 2.4, we have

G(h)φ = |∂x|φ+DN2[h, φ] +DN3[h, h, φ] +DN≥4, (C.53)

where

FDN2[h, φ] =1

∫Rh(ξ − η)φ(η)n2(ξ, η) dη, n2(ξ, η) = ξ · η − |ξ||η|, (C.54)

FDN3[h, h, φ] =1

4π2

∫Rh(ξ − η)h(η − σ)φ(σ)n3(ξ, η, σ) dη,

n3(ξ, η, σ) =|ξ||σ|

2

(|η|+ |ξ + σ − η| − |ξ| − |σ|

),

(C.55)

and, for any t ∈ [0, T ],

‖DN≥4(t)‖HN0 + ‖S(DN≥4)(t)‖HN1 . ε41(1 + t)−5/4. (C.56)

We can now prove Lemma C.5.

Proof of Lemma C.5. We start from the equations (2.24),

∂th = G(h)φ, ∂tφ =∂2xh

(1 + h2x)3/2

− 1

2φ2x +

(G(h)φ+ hxφx)2

2(1 + h2x)

.

Letting U = |∂x|h− i|∂x|1/2φ, it follows that

∂tU − i|∂x|3/2U = QU + CU +R≥4,

where with DN2 = DN2[h, φ], DN3 = DN3[h, h, φ],

QU := |∂x|DN2 +i

2|∂x|1/2

[φ2x − (|∂x|φ)2

],

CU := |∂x|DN3 +3i

2|∂x|1/2(hxxh

2x)− i|∂x|1/2

[|∂x|φ · (DN2 + hxφx)

],

(C.57)

and

R≥4 := |∂x|DN4 − i|∂x|1/2[hxx

( 1

(1 + h2x)3/2

− 1 +3

2h2x

)]− i|∂x|1/2

[ [G(h)φ+ hxφx]2

2(1 + h2x)

− (|∂x|φ)2 + 2|∂x|φ · (DN2 + hxφx)

2

].

Notice also that

h =1

2|∂x|−1(U + U), φ =

i

2|∂x|−1/2(U − U).

The conclusions (C.43) and (C.44) follow from the formula for the quadratic term QU (forthe symbols q++ and q−− it is convenient to symmetrize such that Q++(f, g) = Q++(g, f)and Q−−(f, g) = Q−−(g, f)). The conclusions (C.45)–(C.46) for the cubic components also

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98 ALEXANDRU D. IONESCU AND FABIO PUSATERI

follow from the explicit formulas in (C.57). The symbols m++−,m+++,m−−+,m−−− are linearcombinations, with purely imaginary coefficients, of the symbols

p1(ξ, η, σ) =|ξ|2|σ|

(|η|+ |ξ + σ − η| − |ξ| − |σ|

)|ξ − η||η − σ||σ|1/2

, p2(ξ, η, σ) =|ξ|1/2(ξ − η)2(η − σ)σ

|ξ − η||η − σ||σ|,

p3(ξ, η, σ) =|ξ|1/2|ξ − η|1/2|σ|(|σ| − |η|)

|η − σ||σ|1/2.

(C.58)

More precisely, we have

m++−(ξ, η, σ) =i[−p1(ξ, η, σ) + p1(ξ, η, η − σ) + p1(ξ, ξ − σ, η − σ)]

16

+3i[p2(ξ, η, σ) + p2(ξ, η, η − σ) + p2(ξ, ξ − σ, η − σ)]

16

+i[−p3(ξ, η, σ) + p3(ξ, η, η − σ)− p3(ξ, ξ − σ, η − σ)]

8.

Therefore

m++−(ξ, 0,−ξ) =2i|ξ|3/2

16− 3i|ξ|3/2

16+i|ξ|3/2

8=i|ξ|3/2

16,

as claimed in (C.47).The bounds on the quartic term R≥4 follow easily from the defining formula, Lemma C.2,

(C.51) and Lemma C.6.

C.3.1. The resonant value. We now compute the resonant value c++−(ξ, 0,−ξ), see (8.57),

from the formulas (8.11) and (8.4). Since q+−(0, x) = 0 and m++−(ξ, 0,−ξ) = id1|ξ|3/2 wehave, using (8.52),

ic++−(ξ, 0,−ξ) = 2m++(ξ, 0)q+−(0,−ξ) +m+−(ξ,−ξ)q++(2ξ, ξ)−m+−(ξ, 0)q+−(0, ξ)

− 2m−−(ξ,−ξ)q−−(2ξ, ξ) +m++−(ξ, 0,−ξ)

= m+−(ξ,−ξ)q++(2ξ, ξ)− 2m−−(ξ,−ξ)q−−(2ξ, ξ) + id1|ξ|3/2.

Since q++(2ξ, ξ) = q−−(2ξ, ξ) = i|ξ|3/2√

2/4 and q+−(ξ,−ξ) = 2q−−(ξ,−ξ) = i|ξ|3/2/4, weobtain

c++−(ξ, 0,−ξ) = − q+−(ξ,−ξ)q++(2ξ, ξ)

|ξ|3/2 − |2ξ|3/2 + |ξ|3/2+

2q−−(ξ,−ξ)q−−(2ξ, ξ)

|ξ|3/2 + |2ξ|3/2 + |ξ|3/2+ d1|ξ|3/2

= d2|ξ|3/2,(C.59)

where d2 = −1/16.

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Princeton UniversityE-mail address: [email protected]

Princeton UniversityE-mail address: [email protected]


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